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Solo Hermelin
Solo Hermelin

First part of description of Matrix Calculus at Undergraduate in Science (Math, Physics, Engineering) level. Please send comments and suggestions to solo.hermelin@gmail.com. For more presentations please visit my website at http://www.solohermelin.com.

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1 Matrices I SOLO HERMELIN Updated: 30.03.11http://www.solohermelin.com
2 SOLO Matrices I Table of Content Introduction to Algebra Matrices Vectors and Vector Spaces Matrix Operations with Matrices Domain and Codomain of a Matrix A Transpose AT of a Matrix A Conjugate A* and Conjugate Transpose AH =(A* )T of a Matrix A Sum and Difference of Matrices A and B Multiplication of a Matrix by a Scalar Multiplication of a Matrix by a Matrix Kronecker Multiplication of a Matrix by a Matrix Partition of a Matrix Elementary Operations with a Matrix Rank of a Matrix Equivalence of Two Matrices
3 SOLO Matrices I Table of Content (continue – 1) Matrices Square Matrices Trace of a Square Matrix, Diagonal Square Matrix Identity Matrix, Null Matrix, Triangular Matrices Hessenberg Matrix Toeplitz Matrix, Hankel Matrix Householder Matrix Vandermonde Matrix Hermitian Matrix, Skew-Hermitian Matrix, Unitary Matrix Matrices & Determinants History L, U Factorization of a Square Matrix A by Elementary Operations Invertible Matrices Diagonalization of a Square Matrix A by Elementary Operations
4 SOLO Matrices I Table of Content (continue – 2) Matrices Square Matrices Determinant of a Square Matrix – det A or |A| Eigenvalues and Eigenvectors of Square Matrices Anxn Jordan Normal (Canonical) Form Cayley-Hamilton Theorem Matrix Decompositions Companion Matrix References
5 SOLO Algebra Set and Set Operations A collection of objects sharing a common property is called a Set. We use the notation { }PpropertyhasxxS := We write Sx ∈ S1 is a subset of S if every element of S1 is an element of S { }SxSxxSS ∈→∈∀=⊂ 11 : x is an element of S 1 2 { }elementsno=∅ Null (Empty) set { }2121 : SxorSxxSS ∈∈= Union of sets3 { }2121 : SxandSxxSS ∈∈= Intersection of sets4 { }2121 : SxandSxxSS ∉∈=− Difference of sets5 { }Ω=Ω∈∉= SSandxandSxxS : Complement of S relative to Ω 6 21 SS  1 S 2 S 21 SS  1 S 2 S 21 SS − 1 S 2 S Ω S S
6 SOLO Algebra Set and Set Operations A collection of objects sharing a common property is called a Set. We use the notation { }PpropertyhasxxS := We write Sx ∈ S1 is a subset of S if every element of S1 is an element of S { }SxSxxSS ∈→∈∀=⊂ 11 : x is an element of S 1 2 { }elementsno=∅ Null (Empty) set { }2121 : SxorSxxSS ∈∈= Union of sets3 { }2121 : SxandSxxSS ∈∈= Intersection of sets4 { }2121 : SxandSxxSS ∉∈=− Difference of sets5 { }Ω=Ω∈∉= SSandxandSxxS : Complement of S relative to Ω 6 21 SS  1 S 2 S 21 SS  1 S 2 S 21 SS − 1 S 2 S Ω S S
7 SOLO Algebra Group A nonempty set G is said to be a group if in G there is defined an operation * such that: GbaGba ∈∀∈ ,* Closure1 ( ) ( ) Gcbacbacba ∈∀= ,,**** Associativity2 3 GaaaeeatsGe ∈∀==∈∃ **.., Identity element 4 eabbatsGbGa ==∈∃∈∀ **..,, Inverse element b = a-1 Lemma1: A group G has exactly one identity element Proof: If e and f are both identity elements, then fe ffeef eeffe =⇒    == == ** ** Lemma2: Every element in G has exactly one inverse element Proof: If b and c are both inverse elements of x, then cxebx ** ==   cxbbxb ee **** = *b → → cebe ** = → cb =
8 SOLO Algebra Ring A Ring is a set R equipped with two binary operations +: R ×R→R (called addition), and •: R ×R→R (called multiplication), such that: (R,+) is an Abelian Group with identity element 0: ( ) ( )cbacba ++=++ aaa =+=+ 00 abba +=+ 0..,, =+−=−+∈−∃∈∀ aaaatsRaRa (R,.) is associative ( ) ( )cbacba ••=•• Multiplication distributes over addition: ( ) ( ) ( )cabacba •+•=+• ( ) ( ) ( )cbcacba •+•=•+ RbaRba ∈∀∈+ ,, Closure Associativity Identity element Inverse element Group Properties Abelian Group property
9 SOLO Algebra Field A Field is a Ring satisfying two additional conditions: (1) There also exists an identity e with respect to multiplication, i.e.: aaa =•=• 11 (2) All but the zero element have inverse with respect to multiplication 1..,,0& 111 =•=•∈=∃≠∈∀ −−− aaaatsRabaRa
10 Synthetic Geometry Euclid 300BC Algebras HistorySOLO Extensive Algebra Grassmann 1844 Binary Algebra Boole 1854 Complex Algebra Wessel, Gauss 1798 Spin Algebra Pauli, Dirac 1928 Syncopated Algebra Diophantes 250AD Quaternions Hamilton 1843 Tensor Calculus Ricci 1890 Vector Calculus Gibbs 1881 Clifford Algebra Clifford 1878 Differential Forms E. Cartan 1908 First Printing 1482 http://modelingnts.la.asu.edu/html/evolution.html Geometric Algebra and Calculus Hestenes 1966 Matrix Algebra Cayley 1854 Determinants Sylvester 1878 Analytic Geometry Descartes 1637 Table of Content
11 SOLO Matrices Definitions: Vectors and Vector Spaces Vector: A n-dimensional n-Vector is an ordered set of elements x1, x2,…,xn over a field F. One other way is to define it as Row Matrix or a Column Matrix [ ]n n xxxr x x x c   21 2 1 , =             = we have where T is the Transpose operation.crrc TT == & Scalar: A one-dimensional Vector with its element a real or a complex number. Null Vector: A n-dimensional Vector with all elements equal zero. Equality of two Vectors: niforyxyx ii ,1==⇔= [ ]000, 0 0 0   =             = rc oo
12 SOLO Matrices 12 VECTOR SPACE Given the complex numbers . A Vector Space V (Linear Affine Space) with elements over C if its elements satisfy the following conditions: I. Exists a operation of Addition with the following properties: Commutative (Abelian) Law for Addition1 Associative Law for Addition2 Exists a unique vector3 II. Exists a operation of Multiplication by a Scalar with the following properties: 4 Inverse 5 Associative Law for Multiplication6 Distributive Law for Multiplication7 Commutative Law for Multiplication8 We can write: ( ) ( ) ( ) ( ) ( ) 00101010 3 575 =⋅→==+=⋅+⋅=⋅+ xxxxxxxx ( ) yxyx βαα +=+ ( ) xxx βαβα +=+ ( ) ( )xx βαβα = xx =⋅1 0.. =+∈∃∈∀ yxtsVyVx xx =+ 0 0 ( ) ( )zyxzyx ++=++ xyyx +=+ Vzyx ∈,, C∈γβα ,,
13 SOLO Matrices Linear Dependence and Independence Vectors and Vector Spaces Vectors are said to be Linear Independent if:mvvv ,,, 21  00 212211 =====+++ mmm ifonlyandifvvv αααααα  Vectors are said to be Linear Dependent if :mvvv ,,, 21  0&02211 ≠=+++ imm somevvv αααα  k m ki i iik vv αα / 1           −= ∑ ≠ = If the vectors are Linear Dependent, the vectors whose coefficients αk ≠ 0 in can be obtained as a Linear Combination of other Vectors mvvv ,,, 21  kv 011 =++++ mmkk vvv ααα 
14 SOLO Matrices Linear Dependence and Independence Vectors and Vector Spaces Theorem If Vectors are said to be Linear Independent and vectors are Linear Dependent, than can be expressed as a Unique Linear Combination of . mvvv ,,, 21  121 ,,,, +mm vvvv  mvvv ,,, 21  1+mv Proof 0&0 1112211 ≠=++++ +++ mmmmm vvvv ααααα  since αm+1 = 0 implies mvvv ,,, 21  are Linear Dependent, and this is a contradiction. therefore: ( ) 122111 / ++ +++−= mmmm vvvv αααα  q.e.d. 121 ,,,, +mm vvvv  Linear Dependent implies that exists some (more than one) αi ≠ 0 s.t. To prove Uniqueness suppose that there are two expressions ( ) nivvvv ii tIndependenLinearvvm i iii m i ii m i iim m ,10 ,, 111 1 1 =∀=⇒=−⇒== ∑∑∑ === + γβγβγβ 
15 SOLO Matrices Basis of a Vector Space V Vectors and Vector Spaces A set of Vectors of a n-Vector Space is called a Basis of V if these n Vectors are Linearly Independent and every Vector can be Uniquely expressed as a Linear Combination of those Vectors: nvvv ,,, 21  y ∑= = n i iivy 1 α
16 SOLO Matrices Vectors and Vector Spaces Relation Between Two Bases of a Vector Space V If we have Two Bases of Vectors , we can writenn wwwandvvv ,,,,,, 2121                              =               ⇒        +++= +++= +++= n A nnnn n n nnnnnnn nn nn v v v w w w vvvw vvvw vvvw nxn              2 1 21 22221 11211 2 1 2211 22221212 12121111 ααα ααα ααα ααα ααα ααα In the same way                             =               ⇒        +++= +++= +++= n B nnnn n n nnnnnnn nn nn w w w v v v wwwv wwwv wwwv nxn              2 1 21 22221 11211 2 1 2211 22221212 12121111 βββ βββ βββ βββ βββ βββ Therefore               =               =               n nxnnxn n nxn n v v v AB w w w B v v v  2 1 2 1 2 1 Bnxn is called the Inverse of the Square Matrix Anxn and is written as Anxn -1 .               =               =               n nxnnxn n nxn n w w w BA v v v A w w w  2 1 2 1 2 1 nnxnnxn IBA = nnxnnxn IAB =
17 SOLO Inner Product If V is a complex Vector Space, for the Inner Product (a scalar) < , > between the elements (complex numbers) is defined by: Vzyx ∈∀ ,, * ,, >>=<< xyyx1 Commutative law ><+>>=<+< zxyxzyx ,,,2 Distributive law Cyxyx ∈∀><>=< λλλ ,,3 00,&0, =⇔=><≥>< xxxxx4 Using to we can show that:1 4 ( ) ( ) ( ) ><+><=><+><=>+<=>+< xyxyyxyxyyxxyy ,,,,,, 21 1 * 2 * 1 2 * 21 1 21 ( ) ( ) ><=><=><=>< yxxyxyyx ,,,, * 2 *** 2 λλλλ ( ) >=<>=<⇒><+><=>+>=<< xxxxxx ,000,0,0,00,0, 2 Matrices Vectors and Vector Spaces
18 SOLO Inner Product ( ) ** :, xyyxyx TT ==>< We can define the Inner Product in a Vector Space as Matrices therefore ( ) ∑= =+++>=<⇒             =             = n i iinn nn yxyxyxyxyx y y y y x x x x 1 ** 2 * 21 * 1 2 1 2 1 ,&   Outer Product ( ) [ ]               =             ==>< ** 2 * 1 * 2 * 22 * 12 * 1 * 21 * 11 ** 2 * 1 2 1 * : nnnn n n n n T yxyxyx yxyxyx yxyxyx yyy x x x yxyx       Vectors and Vector Spaces
19 SOLO (Identity) 00 =⇔= xx2 1 Vxx ∈∀≥ 0 (Non-negativity) xx λλ =4 Norm of a Vector .x Vyxyxyxyx ∈∀+≤+≤− ,3 (Triangle Inequalities) Matrices The Norm of a Vector is defined by the following relations: If V is an Inner Product space, than we can induce the norm: [ ] 2/1 , ><= xxx and We can see that 0, 2/1 1 2 2/1 1 *2/1 ≥      =      =>=< ∑∑ == n i i n i ii xxxxxx 1 0,100 2/1 1 2 =⇒=∀=⇒=      = ∑= xnixxx i n i i 2 Vectors and Vector Spaces
20 SOLO Inner Product yxyx ≤>< , Cauchy, Bunyakovsky, Schwarz Inequality known as Schwarz Inequality Let x, y be the elements of an Inner Product space V, than : 0,,,,, 2* ≥><+><+><+>>=<++< yyxyyxxxyxyx ααααα Assuming that (for which the equality holds) we choose: >< >< −= yy yx , , α we have: 0, , , , ,, , ,, , 2 2* ≥>< >< >< + >< ><>< − >< ><>< −>< yy yy yx yy xyyx yy yxyx xx which reduce to: 0 , , , , , , , 222 ≥ >< >< + >< >< − >< >< −>< yy yx yy yx yy yx xx or: ><≥⇔≥><−><>< yxyxyxyyxx ,0,,, 2 q.e.d. Augustin Louis Cauchy )1789-1857( Viktor Yakovlevich Bunyakovsky 1804 - 1889 Hermann Amandus Schwarz 1843 - 1921 MatricesVectors and Vector Spaces 0≠y
21 SOLO Inner Product Cauchy Inequality Let ai, bi (i = 1,…,n) be complex numbers, than :             ≤ ∑∑∑ === n i i n i i n i ii baba 1 2 1 2 2 1 Augustin Louis Cauchy )1789-1857( Viktor Yakovlevich Bunyakovsky 1804 - 1889 Hermann Amandus Schwarz 1843 - 1921 Buniakowsky-Schwarz Inequality ( ) ( ) ( )[ ] ( )[ ]∫∫∫ ≤ dttgdttfdttgtf 22 2 Buniakowsky, V., “Sur quelques inéqualités concernant Les intégrales ordinaires et les intégrales aux différences finite”, Mémoires de l’Acad. de St. Pétersbourg (VII),(1859) Schwarz, H.A., “Über ein die Flächen kleinstein Flächeninhalts betreffendes Problem der Variationsrechnung”, Acta Soc. Scient. Fen., 15, 315-362, (1885) Matrices Vectors and Vector Spaces
22 SOLO Inner Product [ ] 2/1 , ><= xxx Parallelogram law Given an Inner Product space V, than is a norm on V. Moreover for any x,y є X the parallelogram law 2222 22 yxyxyx +=−++ is valid. Proof q.e.d. x y yx + yx − 22 22 22,2,2 ,,,, ,,,, ,, yxyyxx yyxyyxxx yyxyyxxx yxyxyxyxyxyx +>=<+><= ><+><−><−><+ ><+><+><+>=< >−−<+>++=<−++ Matrices Vectors and Vector Spaces
23 SOLO Inner Product Let compute: From this we can see that ><+><= ><−><+><+><− ><+><+><+>=< >−−<−>++=<−−+ xyyx yyxyyxxx yyxyyxxx yxyxyxyxyxyx ,2,2 ,,,, ,,,, ,, 22 ><+><−= ><−><+><−><− ><+><+><−>=< ><−><+><+><− ><+><+><+>=< >−−<−>++=<−−+ xyiyxi yyxyiyxixx yyxyiyxixx yiyixyiyixxx yiyixyiyixxx yixyixyixyixyixyix ,2,2 ,,,, ,,,, ,,,, ,,,, ,, 22 ><=−−++−−+ yxyixiyixiyxyx ,4 2222 *2222 ,4,4 ><>=<=−++−−−+ yxxyyixiyixiyxyx MatricesVectors and Vector Spaces
24 SOLO Norm of a Vector . Matrices Let use the Norm definition to develop the following relations: yxyx yyxxyx yyyxxyxxyxyxyx ,Re2 ,, ,,,,, 22 22 2 ++= +++= +++=++=+ We obtain the Triangle Inequalities yxyxyxyxyx ,2,2 22222 ++≤+≤−+ ( ) ( ) yxyxyxyx ,Re,Im,Re, 22 ≥+=use the fact that: to obtain: use the Scwarz Inequality: ><≥ yxyx , yxyxyxyxyx 22 22222 ++≤+≤−+to obtain: or: ( ) ( )222 yxyxyx +≤+≤− ( ) ( )yxyxyx +≤+≤− Vectors and Vector Spaces x
25 SOLO Norm of a Vector . Matrices Other Definitions of Vector Norms ∑= = n i ixx 1 The following definitions satisfy Vector Norm Properties: 1 2 { }i i xx max= ( ) ( )[ ] ( )[ ] [ ] ∑∑= = ==== n i n j jiij TT xxqxQxxTTxxTxTx 1 1 *2/1* 2/1 ** 2/1 **3 Vectors and Vector Spaces x Return to Table of Content
26 SOLO Matrices Matrix A Matrix A over a field F is a rectangular array of elements in F. If A is over a field of real numbers, A is called a Real Matrix. If A is over a field of complex numbers, A is called a Complex Matrix. A n rows by m columns Matrix A, n x m Matrix, is defined as: [ ]                       ==             = s w o r n r r r ccc aaa aaa aaa A n columnsm m nmnn m m nxm         2 1 21 21 22221 11211 aij (i=1,n,j=1,m) are called the elements of A, and we use also the notation: { }ijaAnxm = Return to Table of Content
27 SOLO Matrices Definitions: Any complex matrix A with n rows (r1, r2,…,rn) and m columns (c1,c2,…,cm) [ ]m n nxm ccc r r r A ,,, 21 2 1   =               = can be considered as a linear function (or mapping or transformation) for a m-dimensional domain to a n-dimensional codomain. ( ) ( ){ }AcodomyAdomxxAyA nxmxnxm ∈⇒∈= 11;: In the same way its conjugate transpose: [ ]H n HH H m H H H mxn rrr c c c A ,,, 21 2 1   =               = is a linear function (or mapping or transformation) for a n-dimensional codomain to a m-dimensional domain. ( ) ( ){ }AcdomxAcodomyyAxA mxnx HH mxn ∈⇒∈= 111111 ;: Operations with Matrices
28 SOLO Matrices Domain and Codomain of a Matrix A The domain of A can be decomposed into orthogonal subspaces: ( ) ( ) ( )ANARAdom H ⊥ ⊕= ( )H AR ( )AN ( )H AN ( )AR xAy = 11 yAx H = ( )Adomxmx ∈1 11mx x ( )Acodomy nx ∈11 1nx yR (AH ) – is the row space of AH (dimension r) N (A) – is the null-space of A (x∈ N (A) ⇔ A x = 0) or the kernel of A (ker (A)) (dimension m-r) The codomain of A (domain of AH ) can be decomposed into orthogonal subspaces: ( ) ( ) ( )H ANARAcodom ⊥ ⊕= R (A) – is the column space of A (dimension r) N (AH ) – is the null-space of AH (dimension n-r) Operations with Matrices Return to Table of Content
29 SOLO Matrices Operations with Matrices The Transpose AT of a Matrix A is obtained by interchanging the rows with the columns. For             = nmnn m m aaa aaa aaa Anxm     21 22221 11211 Transpose AT of a Matrix A the transpose is ( )             == nmmm n n TT aaa aaa aaa AA mxnnxm     21 22212 12111 From the definition it is obvious that (AT )T = A Return to Table of Content
30 SOLO Matrices Operations with Matrices The Conjugate AT of a Matrix A is obtained by tacking the conjugate complex of each of the elements of A. { }* ** 2 * 1 * 2 * 22 * 21 * 1 * 12 * 11 * ij nmnn m m a aaa aaa aaa A nxm =               =     Conjugate A* of a Matrix A the transpose is ( )               == ** 2 * 1 * 2 * 22 * 12 * 1 * 21 * 11 * nmmm n n TH aaa aaa aaa AA nxmmxn     Conjugate Transpose AH =(A* )T of a Matrix A Return to Table of Content
31 SOLO Matrices Operations with Matrices The sum/difference of two matrices A and B of the same dimensions n x m is obtained by adding/subtracting the elements bij to/from elements aij. Sum and Difference of Matrices A and B of the same dimensions n x m { }ijij nmnmnnnn mm mm ba bababa bababa bababa BA nxmnxm ±=             ±±± ±±± ±±± =±     2211 2222222121 1112121111 Given the following transformations 1111 , mxnxmnxmxnxmnx xBzxAy == ( ) 11111 mxnxmnxmmxnxmmxnxmnxnx xBAxBxAzy ±=±=± Return to Table of Content
32 SOLO Matrices Operations with Matrices Multiplication of a Matrix by a Scalar The product of a Matrix by a Scalar is a Matrix in which each Element is multiplied by the Scalar. { }ij nmnn m m a aaa aaa aaa Anxm α ααα ααα ααα α =             =     21 22221 11211 Given the following operations 1111 , mxnxmnxmxnxmnx xAzxAy α== Return to Table of Content
33 SOLO Matrices Operations with Matrices Multiplication of a Matrix by a Matrix Consider the two consecutive transformations: nxp npnn p p mpmm p p nmnn m m C ccc ccc ccc bbb bbb bbb aaa aaa aaa BA mxpnxm =               =                           =             21 22221 11211 21 22221 11211 21 22221 11211 where                             ===             pmpmm p p pxmx m z z z bbb bbb bbb zBx x x x mxp       2 1 21 22221 11211 11 2 1 11 2 1 21 22221 11211 1 2 1 pxmxpnxmmxnxm zBA x x x aaa aaa aaa xAy y y y mnmnn m m nx n =                         ===                  
34 SOLO Matrices Operations with Matrices Multiplication of a Matrix by a Matrix (continue -1) The Multiplication of a Matrix by a Matrix is possible between Matrices in which the number of the columns in the first Matrix is equal to the number of rows in the second Matrix . nxp npnn p p mpmm p p nmnn m m C ccc ccc ccc bbb bbb bbb aaa aaa aaa BA mxpnxm =               =                           =             21 22221 11211 21 22221 11211 21 22221 11211 where ∑= = m j jkijik bac 1 :
35 SOLO Matrices Operations with Matrices Multiplication of a Matrix by a Matrix (continue - 2) CABBCA )()( =Matrix multiplication is associative: Transpose of Matrix Multiplication TTT ABAB =)( Matrix product is compatible with scalar multiplication: ( ) ( )BABAAB ααα ==)( Matrix multiplication is distributive over matrix addition: ( ) CBCACBACABACBA +=++=+ ,)( In general Matrix Multiplication is not Commutative ABAB ≠ Return to Table of Content
36 SOLO Matrices Operations with Matrices Kronecker Multiplication of a Matrix by a Matrix ( ) ( )pmxrnnmnn m m rprr p p nmnn m m BaBaBa BaBaBa BaBaBa bbb bbb bbb aaa aaa aaa BA rxpnxm ⋅⋅             =               ⊗             =⊗             21 22221 11211 21 22221 11211 21 22221 11211 :Leopold Kronecker (1823 –1891) ( ) ( ) ( ) ( ) ( ) ( ) ( )CBACBA BABABA CBCACBA CABACBA ⊗⊗=⊗⊗ ⊗=⊗=⊗ ⊗+⊗=⊗+ ⊗+⊗=+⊗ ααα Properties Return to Table of Content
37 SOLO Matrices Operations with Matrices Partition of a Matrix ( ) ( ) ( ) ( )           =                         = −−− − + +++++ + + pmxqnxpqn pmqxqxp nxm AA AA aaaa aaaa aaaa aaaa A nmnpnpn mqpqpqq qmqpqpq mpp 1221 1211 11 111111 11 111111                     = qpq p aa aa A qxp    1 111 11 : ( )           = + + − qmqp mp aa aa A pmqx    1 111 12 : ( )           = ++ − npn pqq aa aa A xpqn    1 111 21 : ( ) ( )           = + +++ −− nmnp mqpq aa aa A pmxqn    1 111 12 :
38 SOLO Matrices Operations with Matrices Partition of a Matrix (continue) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         ++ ++ =                     = −−−−−−−−−− −−−−−− −−− − −−− − srxpmpmxqnsrpxxpqnxspmpmxqnpxsxpqn srxpmpmqxsrpxqxpxspmpmqxpxsqxp srxpmxspm srpxpxs pmxqnxpqn pmqxqxp mxrnxm BABABABA BABABABA BB BB AA AA BA 2222122121221121 2212121121121111 2221 1211 2221 1211       Return to Table of Content
39 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix j iEE ji cr ↑ ←                 == 100 00 001      ααα AE irα jcEA α The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible) The reverse operation is to multiply the row/column elements by the scalar inverse. ( ) ( ) j iEE ji cr ↑ ←                 == 100 0/10 001 /1/1      ααα ( ) ( ) nrrrr IEEAAEE iiii =⇒= αααα /1/1 ( ) ( ) mcccc IEEAEEA jjjj =⇒= αααα /1/1 1. Multiple the elements of a row/column by a nonzero scalar The reverse operations are written as: ( ) ( ) ( )( ) 1 /1 1 /1 & −− == iiii rrrr EEEE αααα ( ) ( ) ( )( ) 1 /1 1 /1 & −− == jjjj cccc EEEE αααα
40 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix (continue – 1) The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible)                       =→+ 1000 010 0010 0001        α α jij rrrE ←i ←j nrrrrrr IEE jijjij =                       =                                             − =→+→− 1000 0100 0010 0001 1000 010 0010 0001 1000 010 0010 0001                      αα αα The reverse operation is to multiply each element of row i by the scalar (-α) and add to elements of row j AAEE jijjij rrrrrr =→+→− αα                         ++++ =→+ nnnjnin injnijjjiijiij inijiii nji rrr aaaa aaaaaaaa aaaa aaaa AE jij        1 11 1 11111 αααα α ←i ←j 2.a Multiply each element of row i by the scalar α and add to elements of row j                       − =→− 1000 010 0010 0001        α α jij rrrE ←i ←j
41 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix (continue – 2) The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible) ji E jij ccc ↑↑                       =→+ 1000 0100 010 0001        α α ←i ←j ncccccc IEE jijjij =                       =                                             − =→+→− 1000 0100 0010 0001 1000 0100 010 0001 1000 0100 010 0001                      αα αα The reverse operation is to multiply each element of column i by the scalar (-α) and add to elements of column j AEEA jijjij cccccc =→+→− αα                       − =→− 1000 0100 010 0001        α α jij cccE ←i ←j ji aaaaa aaaaa aaaaa aaaaa EA nnninjnin jnjijjjij iniiijiii niji ccc jij ↑↑                         + + + + =→+        α α α α α 1 1 1 111111 ←i ←j 2.b Multiply each element of column i by the scalar α and add to elements of column j
42 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix (continue – 3) The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible)                       =↔ 1000 0010 0100 0001        ji rrE ←i ←j nrrrr IEE jiij =                       =                                             =↔↔ 1000 0100 0010 0001 1000 0010 0100 0001 1000 0010 0100 0001                      The reverse operation is again interchange row j with row i AAEE jiij rrrr =↔↔                         =↔ nnnjnin inijiii jnjjjij nji rr aaaa aaaa aaaa aaaa AE ji        1 1 1 11111 ←i ←j ( ) jiij rrrr EE ↔ − ↔ = 1 3.a Interchange row i with row j
43 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix (continue – 4) The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible) ji E ji rr ↑↑                       =↔ 1000 0010 0100 0001        ncccc IEE ijji =                       =                                             =↔↔ 1000 0100 0010 0001 1000 0010 0100 0001 1000 0010 0100 0001                      The reverse operation is again interchange column j with column i AEEA ijji cccc =↔↔ ji aaaa aaaa aaaa aaaa EA nnninjn jnjijjj iniiiji nij cc ji ↑↑                         =↔        1 1 1 11111 ( ) jiij cccc EE ↔ − ↔ = 1 3.b Interchange column i with column j Return to Table of Content
44 SOLO Matrices Operations with Matrices Rank of a Matrix Given a Matrix Anxm we want, by using Elementary (reversible) Operations to reduce it to a Main Diagonal Unit Matrix and zeros in all other positions.             = nmnn m m aaa aaa aaa Anxm     21 22221 11211 Assume that a11 ≠ 0. If this is not the case interchange the first row/column (a Elementary operation) until this is satisfied. Divide the elements of the first row by a11. For i=2,n multiply the first row by (–ai1/a11) and add to i row (a Elementary operation) to obtain:                       −− −− = m n nm n n mm m a a a aa a a a a a a aa a a a a a a a AE nxm 1 11 1 12 11 1 2 1 11 21 212 11 21 22 11 1 11 12 1 0 0 1    
45 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 1) Repeat this procedure for second column (starting at the new a22), third column (starting at the new a33), and so on, as long as we ca obtain non-zero elements on the main diagonal, using the rows bellow. At the end we obtain: r a aa aaa AEEE nm mr mr rowrowrrow nxm ↑                       = 0000 0000 '100 ''10 '''1 22 1112 1_2__         ←r Define the multiplications of Elementary Operations as: 1_2__: rowrowrrow EEEP = Those Elementary Operations can be reversed in opposite order to obtain: ( ) ( ) ( ) 1 _ 1 2_ 1 1_ 1 : −−−− = rrowrowrow EEEP  nIPP =−1
46 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 2) Now use column operation starting with the first column in order to nullify all the elements above the Main Unit Diagonal: ( ) ( ) ( ) ( )         =                       = −−− − rmxrnxrrn rmrxr rcccrowrowrrow I EEEAEEE nxm 00 0 0000 0000 0100 0010 0001 _2_1_1_2__         Define the multiplications of Elementary Operations as: rccc EEEQ _2_1_: = Those Elementary Operations can be reversed in opposite order to obtain: ( ) ( ) ( ) 1 1_ 1 2_ 1 _ 1 : −−−− = ccrc EEEQ  mIQQ =−1
47 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 3) We obtained:  1111 00 0 00 0 −−−−       =⇒      = Q I PQQAPP I QPA r I I r m nxm n nxm  The maximum number of Linearly Independent Rows of A = r 11 00 0 −−       = Q I PA r nxm From the relation we can see that the maximum number of Linearly Independent Rows and the maximum number of Linearly Independent Columns of Matrix PAQ is r.       = 00 0rI QPAnxm       =               =      = −− −− −− − 0000 0 00 0 12 1 11 1 22 1 21 1 12 1 11 1 1 QQ QQ QQI Q I PA rr nxm Since the maximum number of Linearly Independent Rows of Matrix PA is also r. But the Elementary Operations P are not changing the number of Linearly Independent Rows of A, therefore: The maximum number of Linearly Independent Columns of A = r         =              =      = − − −− −− − 0 0 00 0 00 0 21 1 11 1 22 1 21 1 12 1 11 1 1 P PI PP PPI PQA rr nxm Since the maximum number of Linearly Independent Columns of Matrix A Q is also r. But the Elementary Operations Q are not changing the number of Linearly Independent Columns of A, therefore:
48 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 4) We obtained:       = 00 0rI QPAnxm The maximum number of Linearly Independent Rows of Anxm = The maximum number of Linearly Independent Columns of Anxm = r ≤ min (m,n) := Rank of Matrix Anxm 11 00 0 −−       = Q I PA r nxm ( ) ( ) ( )TrT mxn TT P I QAAnxm 11 00 0 −−       == Since in the Transpose of A we interchanged the columns with the rows of A: nxmmxn T ARankARank =
49 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 5) Proof ( ) ( ) mxpmxp mxp BRankBARank ARankBARank nxm nxmnxm ≤ ≤ Rank of A B: ( )mnrARank nxm ,min≤=Assume:       =               =      =⇒      = −− −− −− − 0000 0 00 0 00 0 12 1 11 1 22 1 21 1 12 1 11 1 1 QQ QQ QQI Q I PA I QPA rrr nxmnxm ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         =                 = −−− − −−− − −−− − −− rpxrnxrrn rprxrxr rpxrmxrrm rprxrxr rmxrnxrrn rmrxrxr mxp HH BB BBQQ BPAnxm 0000 1211 1221 121112 1 11 1 Therefore (P A B) has at most r nonzero rows: ARankrABRankPABRank rNonsingulaP =≤= Since ( ) ( ) BRankBRankABRankABRankABRankABAB TTTTTTT =≤==⇒= q.e.d.
50 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 6) ( ) ( ) nxnnxnnxn nxnnxnnxn BRankARankmBARank BRankARankBARank nxn nxn +≤+ +≤+ If A and B are Square nxn Matrices then: [3] K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967, p.104 ( ) ( )nxpmxnmxnnxpmxn BRankARankBARanknBRankARank nxp ,min≤≤−+ Sylvester’s Inequality: James Joseph Sylvester (1814 – 1887) [4] T. Kailath, “Linear Systems”, Prentice Hall, Inc., 1980, p.654 Return to Table of Content
51 SOLO Matrices Operations with Matrices Equivalence of Two Matrices Proof Two Matrices Anxm and Bnxm are said to be Equivalent, if and only if there exist a Nonsingular Matrix Pnxn and a Nonsingular Matrix Qmxm such that A=P B Q. This is the same to saying that A and B are Equivalent if and only if they have the same rank. Since A and B have the same rank r, we can write: T I SBH I GA rr       =      = 00 0 , 00 0 where G,H, S, T are square invertible matrices. q.e.d. QBPHTBSGATBS I QP r ==⇒=      −−−−  1111 00 0 P and Q are square invertible matrices since ( ) ( ) THHTQGSSGPHTQSGP 1111111111 ,:&: −−−−−−−−−− ====⇒== Return to Table of Content
52 SOLO Matrices Square Matrices In a Square Matrix Number of Rows = Number of Columns = n             = nnnn n n aaa aaa aaa Anxn     21 22221 11211 Trace of a Square Matrix ∑= == n i iiaAtrAoftrace nxnnxn 1 Diagonal Square Matrix { }ijij nn a a a a Dnxn δ=             =     00 00 00 22 11 Return to Table of Content
53 SOLO Matrices Square Matrices Identity Matrix Triangular Matrices { }ijnnxn II δ=             == 100 010 001     A Matrix whose elements below or above the main diagonal are all zero is called a Triangular Matrix             = nnnn aaa aa a Lnxn     21 2221 11 0 00 nxnnxnnxnnxnnxn AAIIA == Null Matrix { }0=nxn O nxnnxnnxnnxnnxn OOIIO == Upper Triangular Matrix Lower Triangular Matrix             = nn n n a aa aaa Unxn     00 0 222 11211 Return to Table of Content
54 SOLO Matrices Square Matrices Hessenberg Matrix An Upper Hessenberg Matrix has zero entries below the first subdiagonal: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                         = − −− −− −− −− nnnn nnn nnn nnn nnn H aa aaaa aaaaa aaaaaa aaaaaa U nxn 1 4142443 313233332 21222232211 11121131211 0000 00 0       An Lower Hessenberg Matrix has zero entries below the first superdiagonal: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                         = −−−−− −−−− nnnnnn nnnnnn nnnn H aaaaa aaaaa aaaa aaaa aaa aa L nxn        4321 141312111 42322212 34333231 232221 1211 0 0 00 000 A Hessenberg Matrix is an “almost” Triangular Matrix. Return to Table of Content
55 SOLO Matrices Square Matrices Toeplitz Matrix A Toeplitz Matrix or a “Diagonal-constant Matrix”, named after Otto Toeplitz, is a Matrix in which each descending Diagonal from left to right is constant. Otto Toeplitz (1881 – 1940)                    = − − −− − +−−− 0121 101 21 012 101 1210 aaaa aaa aa aaa aaa aaaa T n n nxn       Hankel Matrix A Hankel Matrix is closed related to a Toeplitz Matrix (a Hankel Matrix is an upside-down Toeplitz Matrix), named after Hermann Hankel, is a Matrix in which each uprising Diagonal from left to right is constant.                       = −+−+− +− −− −−− +−−− nnn n n nxn aaa a aa aaa aaaa H 2121 12 32 321 1210        Hermann Hankel (1839 – 1873) Return to Table of Content
56 SOLO Matrices Householder Matrix nˆ ( )xnn T  ˆˆ ( )xnn T  ˆˆ x  'x  O A We want to compute the reflection of over a plane defined by the normal ( )1ˆˆˆ =nnn T x  From the Figure we can see that: ( ) ( ) xHxnnIxnnxx TT  =−=−= ˆˆ2ˆˆ2' 1ˆˆˆˆ2: =−= nnnnIH TT We can see that H is symmetric: ( ) HnnInnIH TTTT =−=−= ˆˆ2ˆˆ2 In fact H is also a rotation of around OA so it must be orthogonal, i.e. HT H=H HT =I. x  ( ) ( )  InnnnnnInnInnIHHHH TTTTTT =+−=−−== ˆˆˆˆ4ˆˆ4ˆˆ2ˆˆ2 1 Alston Scott Householder 1904-1993 Square Matrices Return to Table of Content
57 SOLO Matrices Square Matrices Vandermonde Matrix ( )                 = −−− 11 2 1 1 22 2 2 1 21 21 111 ,,, n n nn n n n xxx xxx xxx xxxVnxn       Vandermonde Matrix is a nxn Matrix that has in its j row the entries x1 j-1 x2 j-1 … xn j-1 Alexandre-Théophile Vandermonde 1735 - 1796 Return to Table of Content
58 SOLO Hermitian = Symmetric if A has real components Hermitian Matrix: AH = A, Symmetric Matrix: AT = A Matrices Pease, “Methods of Matrix Algebra”, Mathematics in Science and Engineering Vol.16, Academic Press 1965 Definitions: Adjoint Operation (H): AH = (A*)T (* is complex conjugate and T is transpose of the matrix) Skew-Hermitian = Anti-Symmetric if A has real components. Skew-Hermitian: AH = -A, Anti-Symmetric Matrix: AT =-A Unitary Matrix: UH = U-1, Orthonormal Matix: OT = O-1 Unitary = Orthonormal if A has real components. Charles Hermite 1822 - 1901 Square Matrices Hermitian Matrix, Skew-Hermitian Matrix, Unitary Matrix Return to Table of Content
59 SOLO Matrices Square Matrices Singular, Non-singular and Inverse of a Non-singular Square Matrix Anxn We obtained:       = 00 0rI QPAnxn 11 00 0 −−       = Q I PA r nxn Singular Square Matrix Anxn: r < n Only r rows/columns of A are Linearly Independent Non-singular Square Matrix Anxn: r = n The n rows/columns of A are Linearly Independent For a Non-singular Matrix (r=n): n I n IQQQPPQAPQQPQIPA n nxn ===⇒== −−−−−−− 1111111  and:  n I n IPPPQQPPQAQPQIPA n nxn ===⇒== −−−−−−− 1111111 The Matrix (Q P) is the Inverse of the Non-singular Matrix A: PQAnxn = −1 This result explains the Gauss–Jordan elimination algorithm that can be used to determine whether a given square matrix is invertible and to find the inverse Return to Table of Content
60 SOLO Matrices Invertible Matrices Matrix Inversion • Gauss–Jordan elimination is an algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. • An alternative is the LU decomposition which generates an upper and a lower triangular matrices which are easier to invert. • For special purposes, it may be convenient to invert matrices by treating mxn- by-mxn matrices as m-by-m matrices of n-by-n matrices, and applying one or another formula recursively (other sized matrices can be padded out with dummy rows and columns). • For other purposes, a variant of Newton's method may be convenient (particularly when dealing with families of related matrices, so inverses of earlier matrices can be used to seed generating inverses of later matrices). Square Matrices
61 SOLO Matrices Invertible Matrices Square Matrices Gaussian elimination, which first appeared in the text Nine Chapters on the Mathematical Art written in 200 BC, was used by Gauss in his work which studied the orbit of the asteroid Pallas. Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix. Sketch of the orbits of Ceres and Pallas, by Gauss http://www.math.rutgers.edu/~cherlin/History/Papers1999/ weiss.html Gauss published his methods in 1809 as "Theoria motus corporum coelestium in sectionibus conicus solem ambientium," or, "Theory of the motion of heavenly bodies moving about the sun in conic sections."
62 SOLO Matrices Invertible Matrices Gauss-Jordan elimination In Linear Algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is variation of Gaussian elimination. Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards. Matrices containing zeros below each pivot are said to be in row echelon form. Gauss–Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form. Every matrix has a reduced row echelon form, and Gauss–Jordan elimination is guaranteed to find it. Carl Friedrich Gauss (1777–1855) Wilhelm Jordan ( 1842–1899) See example Square Matrices
63 SOLO Matrices Invertible Matrices Gauss-Jordan elimination If the original square matrix, A, is given by the following expression:           − −− − = 210 121 012 33xA Then, after augmenting A Matrix by the Identity Matrix, the following is obtained: [ ]           − −− − = 100210 010121 001012 IA Perform the following: 1. row1 + row2 →row1 equivalent with left multiplication by           =→+ 100 010 011 121 rrrE [ ]           − −− − =→+ 100210 010121 011111 121 IAE rrr Square Matrices
64 SOLO Matrices Invertible Matrices Gauss-Jordan elimination 2. row1 + row2 →row2 equivalent with left multiplication by           =→+ 100 011 001 221 rrrE [ ]           − −− − =→+ 100210 010121 011111 121 IAE rrr [ ]           − − − =→+→+ 100210 021230 011111 121221 IAEE rrrrrr 3. (1/3) row2 →row2 equivalent with left multiplication by           = → 100 03/10 001 22 3 1 rr E [ ]             − − − =→+→+ → 100210 0 3 2 3 1 3 2 10 011111 121221 22 3 1 IAEEE rrrrrr rr 1. row1 + row2 →row1 Square Matrices
65 SOLO Matrices Invertible Matrices Gauss-Jordan elimination 3. (1/3) row2 →row2           =→+ 110 010 001 332 rrrE [ ]             − − − =→+→+ → 100210 0 3 2 3 1 3 2 10 011111 121221 22 3 1 IAEEE rrrrrr rr 4. row2+row3 →row3 equivalent with left multiplication by [ ]                 − − =→+→+ → →+ 1 3 2 3 1 3 4 00 0 3 2 3 1 3 2 10 011111 121221 22 332 3 1 IAEEEE rrrrrr rr rrr 5. row1-row2 →row1 equivalent with left multiplication by           − =→− 100 010 011 121 rrrE [ ]                   − − =→+→+ → →+→− 1 3 2 3 1 3 4 00 0 3 2 3 1 3 2 10 0 3 1 3 2 3 1 01 121221 22 332121 3 1 IAEEEEE rrrrrr rr rrrrrr Square Matrices
66 SOLO Matrices Invertible Matrices Gauss-Jordan elimination 5. row1-row2 →row1               = → 4 3 00 010 001 33 4 3 rr E [ ]                   − − =→+→+ → →+→− 1 3 2 3 1 3 4 00 0 3 2 3 1 3 2 10 0 3 1 3 2 3 1 01 121221 22 332121 3 1 IAEEEEE rrrrrr rr rrrrrr 6. (4/3) row3 →row3 equivalent with left multiplication by [ ]                   − − =→+→+ → →+→− → 4 3 2 1 4 1 100 0 3 2 3 1 3 2 10 0 3 1 3 2 3 1 01 121221 22 332121 33 3 1 4 3 IAEEEEEE rrrrrr rr rrrrrr rr 7. (1/3) row3+row1 →row1 equivalent with left multiplication by               = →+ 100 010 3 1 01 113 3 1 rrr E [ ]                   −=→+→+ → →+→− →→+ 4 3 2 1 4 1 100 0 3 2 3 1 3 2 10 4 1 2 1 4 3 001 121221 22 332121 33113 3 1 4 3 3 1 IAEEEEEEE rrrrrr rr rrrrrr rrrrr Square Matrices
67 SOLO Matrices Invertible Matrices Gauss-Jordan elimination 7. (1/3) row3+row1 →row1             = →+ 100 3 2 10 001 223 3 2 rrr E [ ]                   −=→+→+ → →+→− →→+ 4 3 2 1 4 1 100 0 3 2 3 1 3 2 10 4 1 2 1 4 3 001 121221 22 332121 33113 3 1 4 3 3 1 IAEEEEEEE rrrrrr rr rrrrrr rrrrr 8. (2/3) row3+row2 →row2 equivalent with left multiplication by [ ] [ ]BIIAEEEEEEEE B rrrrrr rr rrrrrr rrrrrrrr =                   =→+→+ → →+→− →→+→+ 4 3 2 1 4 1 100 2 1 1 2 1 010 4 1 2 1 4 3 001 121221 22 332121 33113223 3 1 4 3 3 1 3 2    We found [ ] [ ] 1− =⇒=⇒= ABIABBIIAB 1 3 1 4 3 3 1 3 2 4 3 2 1 4 1 2 1 1 2 1 4 1 2 1 4 3 : 121221 22 332121 33113223 − →+→+ → →+→− →→+→+ =                   == AEEEEEEEEB rrrrrr rr rrrrrr rrrrrrrr [ ] [ ]11 −− =− AIIAAinationlimeJordanGaussTherefore Square Matrices Return to Table of Content
68 The first to use the term 'matrix' was Sylvester in 1850. Sylvester defined a matrix to be an oblong arrangement of terms and saw it as something which led to various determinants from square arrays contained within it. After leaving America and returning to England in 1851, Sylvester became a lawyer and met Cayley, a fellow lawyer who shared his interest in mathematics. Cayley quickly saw the significance of the matrix concept and by 1853 Cayley had published a note giving, for the first time, the inverse of a matrix. Arthur Cayley 1821 - 1895 Cayley in 1858 published “Memoir on the Theory of Matrices” which is remarkable for containing the first abstract definition of a matrix. He shows that the coefficient arrays studied earlier for quadratic forms and for linear transformations are special cases of his general concept. Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses. He gave an explicit construction of the inverse of a matrix in terms of the determinant of the matrix. Cayley also proved that, in the case of 2 2 matrices, that a matrix satisfies its own characteristic equation. James Joseph Sylvester 1814 - 1897 http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86 Return to Table of Content
69 SOLO Matrices Square Matrices L, U Factorization of a Square Matrix A by Elementary Operations Given a Square Matrix Number of Rows = Number of Columns = n             = nnnn n n aaa aaa aaa Anxn     21 22221 11211 Consider the following Simple Operations on the rows/columns of A to obtain a U Triangular Matrix (all elements bellow the Main Diagonal are 0) : 1. Multiple the elements of a row/column by a nonzero scalar 2. Multiply each element of row i by the scalar α and add to elements of row j j iEE ji cr ↑ ←                 == 100 00 001      αααAE irα jcEA α                       =↔+ 1000 010 0010 0001        α α jij rrrE AE jij rrr →+α L,U factorization was proposed by Heinz Rutishauser in 1955.
70 SOLO Matrices Square Matrices L, U Factorization of a Matrix A by Elementary Operations Given a Square Matrix Number of Rows = Number of Columns = n for example:           − −− − = 210 121 012 33xA Consider the following Simple Operations on the rows/columns of A to obtain a U1 Triangular Matrix (all elements bellow the Main Diagonal are 0) :           = →+ 13/20 010 001 332 3 2 rrr E             − − − = →+ 210 1 2 3 0 012 221 2 1 AE rrr           = →+ 100 012/1 001 221 2 1 rrr E 1 2 1 3 2 3 4 00 1 2 3 0 012 221332 UAEE rrrrrr =                 − − = →+→+ 1. (1/2) row1+row2 →row1 equivalent with left multiplication by 2. (2/3) row2 + row3 →row3 equivalent with left multiplication by
71 SOLO Matrices Square Matrices L, U Factorization of a Matrix A by Elementary Operations           =                     = →→+ 13/23/1 012/1 001 100 012/1 001 13/20 010 001 11221 2 1 2 1 rrrrr EE 1 2 1 3 2 3 4 00 1 2 3 0 012 221332 UAEE rrrrrr =                 − − = →+→+ we found: To Undo the Simple Operations and to obtain again A, let perform:           − = →+ 13/20 010 001 332 3 2 rrr E we can see that:           =                     − = →+→+ 100 010 001 13/20 010 001 13/20 010 001 332332 3 2 3 2 rrrrrr EE 332 3 2 rrr E →+− is the Inverse Operation to and we write 1 3 2 3 2 332332 − →+→+−         = rrrrrr EE 332 3 2 rrr E →+           −= →+− 100 012/1 001 221 2 1 rrr E           =                     −= →+→+− 100 010 001 100 012/1 001 100 012/1 001 221221 2 1 2 1 rrrrrr EE 221 2 1 rrr E →+− is the Inverse Operation to and we write 1 2 1 2 1 221221 − →+→+−         = rrrrrr EE 221 2 1 rrr E →+ 1. (-2/3) row2 + row3 →row3 equivalent with left multiplication by 2. (-1/2) row1+row2 →row1 equivalent with left multiplication by
72 SOLO Matrices Square Matrices L, U Factorization of a Matrix A by Elementary Operations LEE rrrrrr =           − −=           −          −= →+−→+− 13/20 012/1 001 13/20 010 001 100 012/1 001 221332 2 1 3 2 AUEEAEEEE rrrrrrrrrrrrrrrrrr =        =                →+−→+−→+→+→+−→+− 1 3 2 2 1 2 1 3 2 3 2 2 1 332221221332332221 we found: Therefore we obtained an L U factorization of the Square Matrix A: AUL =           − −− − =                 − −           − −= 210 121 012 3 4 00 1 2 3 0 012 13/20 012/1 001 1 We can have 1 on the diagonal of U Matrix, by introducing the Diagonal Matrix D: UDLA =           − −                     − −=           − −− − = 100 2/310 02/11 4/300 03/20 002/1 13/20 012/1 001 210 121 012 Return to Table of Content
73 SOLO Matrices Square Matrices Diagonalization of a Square Matrix A by Elementary Operations we found: 1 2 1 3 2 3 4 00 1 2 3 0 012 221332 UAEE rrrrrr =                 − − = →+→+ 1. (3/2) row2 + row1 →row1 equivalent with left multiplication by 2. (4/3) row3 + row2 →row2 and (9/8) row3 + row1 →row1 equivalent with left multiplication by               = →+ 100 010 0 2 3 1 112 2 3 rrr E                   − − = →+→+→+ 3 4 00 1 2 3 0 2 3 02 221332112 2 1 3 2 2 3 AEEE rrrrrrrrr               = →+→+ 100 010 8 9 3 4 1 223113 3 4 8 9 rrrrrr EEDAEAEEEEE rrrrrrrrrrrrrrr =                 == →+→+→+→+→+ 3 4 00 0 2 3 0 002 221332112223113 2 1 3 2 2 3 3 4 8 9 Return to Table of Content
74 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             = To each Matrix A we associate a scalar called Determinant; i.e. det A or |A| defined by the following 4 properties: 1 The Determinant of the Identity Matrix In is 1. If the Matrix A has two identical rows/columns the Determinant of A is zero.2 0det 1 =                       nr r r r    α α [ ] 0det 1 =ncccc  αα ←i row ↑ i column 1 1000 0100 0010 0001 detdet =                     =       nI
75 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| To each Matrix A we associate a scalar called Determinant; i.e. det A or |A| defined by the following 4 properties: 3 If each element of a row/column of the Matrix A is the sum of two terms, the Determinant of A is the sum of the two Determinants formed by the separation of the terms                 +                 =                 + n k n k n kk r r r r r r r rr r       'detdet'det 111 [ ] [ ] [ ]nknknkk cccccccccc  'detdet'det 111 +=+ [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
76 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 4 If the elements of a row/column of the Matrix A have a common factor λ than the Determinant of A is equal to the product of λ and the Determinant of the Matrix obtained by dividing the previous row/column by λ.                 =                 nknn knkk n nknn knkk n aaa aaa aaa aaa aaa aaa           21 21 11211 21 21 11211 detdet λλλλ [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
77 SOLO Matrices & Determinants History http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86 The idea of a determinant appeared in Japan and Europe at almost exactly the same time although Seki in Japan certainly published first. In 1683 Seki wrote “Method of solving the dissimulated problems “ which contains matrix methods written as tables. Without having any word which corresponds to 'determinant' Seki still introduced determinants and gave general methods for calculating them based on examples. Using his 'determinants' Seki was able to find determinants of 2x2, 3x3, 4x4 and 5x5 matrices and applied them to solving equations but not systems of linear equations. Takakazu Shinsuke Seki 1642 - 1708Rather remarkably the first appearance of a determinant in Europe appeared in exactly the same year 1683. In that year Leibniz wrote to de l'Hôpital. He explained that the system of equations 10 + 11x + 12y = 0 20 + 21x + 22y = 0 30 + 31x + 32y = 0 had a solution because 302112322011312210312012302211322110 ⋅⋅+⋅⋅+⋅⋅=⋅⋅+⋅⋅+⋅⋅ which is exactly the condition that the coefficient matrix has determinant 0. Gottfried Wilhelm von Leibniz 1646 - 1716
78 SOLO Matrices & Determinants History http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86 Leibniz used the word 'resultant' for certain combinatorial sums of terms of a determinant. He proved various results on resultants including what is essentially Cramer's rule. He also knew that a determinant could be expanded using any column - what is now called the Laplace expansion. As well as studying coefficient systems of equations which led him to determinants, Leibniz also studied coefficient systems of quadratic forms which led naturally towards matrix theory. Gottfried Wilhelm von Leibniz 1646 - 1716 Gabriel Cramer (1704-1752) In the 1730's Maclaurin wrote Treatise of algebra although it was not published until 1748, two years after his death. It contains the first published results on determinants proving Cramer's rule for 2x2 and 3x3 systems and indicating how the 4x4 case would work. Cramer gave the general rule for n n systems in a paper Introduction to the analysis of algebraic curves (1750). It arose out of a desire to find the equation of a plane curve passing through a number of given points. Cramer does go on to explain precisely how one calculates these terms as products of certain coefficients in the equations and how one determines the sign. He also says how the n numerators of the fractions can be found by replacing certain coefficients in this calculation by constant terms of the system. Colin Maclaurin 1698 - 1746
79 An axiomatic definition of a determinant was used by Weierstrass in his lectures and, after his death, it was published in 1903 in the note ‘On Determinant Theory‘. In the same year Kronecker's lectures on determinants were also published, again after his death. With these two publications the modern theory of determinants was in place but matrix theory took slightly longer to become a fully accepted theory. Karl Theodor Wilhelm Weierstrass 1815 - 1897 Leopold Kronecker 1823 - 1891 Determinant Weirstrass Definition of Determinant of a nxn Matrix A: (1)det (A) is linear in the rows of A (2) Interchanging two rows change the sign of det (A) (3) det (In) = 1 For each positive integer n, there is exactly one function with these three properties. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86 http://www.sandgquinn.org/stonehill/MA251/notes/Weierstrass.pdf
80 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 5 If in a Matrix Determinant we interchange two rows/columns the sign of the Determinant will change. Proof [ ]nji cccc 1detgiven ( ) [ ] ( ) [ ] ( ) [ ]nji by nii njiji cccccccc cccccc       1 20 1 3 1 2 detdet det0 += ++= [ ] [ ] ( )     20 11 detdet by njjnij cccccccc ++ therefore [ ] [ ]nijnji cccccccc  11 detdet −= q.e.d. [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
81 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 6 The Matrix Determinant is unchanged if we add to a row/column any linear combination of the other rows/columns. Proof [ ]nji cccc 1detgiven q.e.d. [ ] [ ] ( ) [ ]ni ij j by njjj nin ij j jji ccccccc ccccccc       1 20 1 11 detdet detdet =+ =           + ∑ ∑ ≠ ≠ λ λ [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
82 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 7 If a row/column is a Linear Combination of other rows/columns the Determinant is zero. Proof q.e.d. [ ] ( ) 0detdet 20 11 ==           ∑∑ ≠≠ ij j by njjjn ij j jj ccccccc     λλ [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
83 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 8 Leibniz formula for determinants ( ) ( )n n i n iii i n iii i nikii L nnnknn nk nk iiinPermutatioL aaa aaaa aaaa aaaa A kk k nn n nk ,,, 1detdet 21 1 21 222221 111211 1 11 11 1         = −=             = ∑ ∑ ∑ − −≠ ≠ The meaning of this equation is that in the product there are no two elements of the same row or the same column, and the sign of the product is a function of the position of each element in the Matrix. The sign of each element, in the product, is given by { } ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               −−−−− −−−+− −−+−+ =−= +++++ ++ ++ + nnknnnn nk nk ji ijasign 11111 11 11 1 321 22 11     Gottfried Wilhelm Leibniz (1646 – 1716) [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
84 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof 8 ( ) ( )n n i n iii i n iii i nikii L nnnknn nk nk iiinPermutatioL aaa aaaa aaaa aaaa A kk k nn n nk ,,, 1detdet 21 1 21 222221 111211 1 11 11 1         = −=             = ∑ ∑ ∑ − −≠ ≠ From Properties (3) and (4) of the Determinant: ( ) ( ) ( ) ( ) [ ]010:detdetdetdet 1 321 4,3 1 2 1 4,3 21 222221 111211 1 2 2 1 21 1 1 1        =                   =               =             = ∑ ∑∑ == i n i n i n i i ii n i n i i nnnknn nk nk ewhere r r e e aa r r e a aaaa aaaa aaaa A ↑ i column ↑ 1st row coeff 2nd row Coeff ↓ From Properties (2) if two rows are identical the determinant is zero, therefore, in the summation of i2 we can delete the case i2=i1. [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
85 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof (continue 1) 8 ( ) ( )n n i n iii i n iii i nikii L nnnknn nk nk iiinPermutatioL aaa aaaa aaaa aaaa A kk k nn n nk ,,, 1detdet 21 1 21 222221 111211 1 11 11 1         = −=             = ∑ ∑ ∑ − −≠ ≠ From Properties (2),(3) and (4) of the Determinant: ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ∑∑ ∑ ∑ ∑∑ ≠ ≠ == −               = =                   =               =             = n i n ii i n iiii i i i i niii i n i n i n i i ii n i n i i nnnknn nk nk nn n n n e e e aaa ewhere r r e e aa r r e a aaaa aaaa aaaa A 1 12 2 121 2 1 21 1 2 2 1 21 1 1 1 ,,, 21 4,3,2 1 321 4,3 1 2 1 4,3 21 222221 111211 det 010:detdetdetdet           ↑ i column [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
86 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof (continue 1) q.e.d. 8 ( ) ( )n n i n iii i n iii i nikii L nnnknn nk nk iiinPermutatioL aaa aaaa aaaa aaaa A kk k nn n nk ,,, 1detdet 21 1 21 222221 111211 1 11 11 1         = −=             = ∑ ∑ ∑ − −≠ ≠ Let interchange the position of the rows to obtain a Unit Matrix, where, according with Property (5), each interchange will cause a change in determinant sign. We also use Property (1) that the determinant of the Unit Matrix is 1: ( )∑ ∑ ∑ − −≠ ≠ −=             = n i n iii i n iii i nikii L nnnknn nk nk kk k nn n nk aaa aaaa aaaa aaaa A 1 11 11 1 ,, ,, 1 21 222221 111211 1detdet        ( ) ( ) ( )L n L i i i e e e e e e n 1det1det 1 2 1 2 1 −=             −=                where L is the Number of Permutations necessary to go from (i1,i2,…,in) to (1,2,…,n) [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
87 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 9 A Determinant can be expanded along a row or column using Laplace's Formula: ( )∑∑ = + = −== n k ki ki ik n k kiik MaCaA 1 , 1 , 1det where the Ci,k represents the i,k element of the matrix cofactors, i.e. Ci,k is ( − 1)i + k times the minor Mi,k, which is the determinant of the matrix that results from A by removing the i-th row and the k-th column, and n is the length of the matrix. Pierre-Simon, marquis de Laplace 1749 - 1827 ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )                         = +− ++++−++ +− −+−−−−− +− nnknnkknn nikikikii inkiikkii nikikikii nkkk ki aaaaa aaaaa aaaaa aaaaa aaaaa M        111 11111111 111 11111111 11111111 , det [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
88 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 9 Laplace's Formula: ( ) ∑∑∑ == + = =−== n j ijji n j ji ji ij n j jiij CaMaCaA 1 , 1 , 1 , 1det Proof ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑= +− +− +−                     =             = n k nnknkknn nkkk nkkk ik nnnknn nk nk aaaaa aaaaa aaaaa a aaaa aaaa aaaa A 1 1111 21121221 11111111 4,3 21 222221 111211 00100 detdetdet           From Properties (3) and (4) of the Determinant, using Row summation: From Properties (3) and (5) of the Determinant: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                 +               −++               ⋅=                     +− +− +− + +− +− +− +− +− +− nnkknn nkk nkk ki nnknkknn nkkk nkkk nnknkknn nkkk nkkk aaaa aaaa aaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa 1111 2111221 1111111 1112 21121222 11111112 1111 21121221 11111111 det1det0 00100 det ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ki ki nnkknn nkk nkk ki nnknkknn nkkk nkkk M aaaa aaaa aaaa aaaaa aaaaa aaaaa , 1111 2111221 1111111 1111 21121221 111111111 1det1det0 + +− +− +− + +− +− −+− −=               −=               ⋅+          [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
89 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 9 Laplace's Formula: ( ) ∑∑∑ == + = =−== n j ijji n j ji ji ij n j jiij CaMaCaA 1 , 1 , 1 , 1det ( )∑∑ = + = −== n j ji ji ij n j jiij MaCaA 1 , 1 , 1det Proof (continue 1) Therefore the minor Mi,k, which is the determinant of the matrix that results from A by removing the i-th row and the k-th column. We obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ki ki nnknkknn nkkk nkkk ki nnknkknn nkkk nkkk ki M aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa C , 1111 21121221 11111111 1111 21121221 11111111 , 1: 00100 det1 00100 det: + +− +− +− + +− +− +− −=                     −=                     =             q.e.d. In the same way we can use Column summation to obtain ( )∑∑ = + = −== n j ij ji ji n j ijji MaCaA 1 , 1 , 1det [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
90 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 10 A-1 the Inverse of Matrix A with det A ≠ 0 is unique and given by:               ==− nnninn ni ni CCCC CCCC CCCC Aadjwhere A Aadj A ,,,2,1 2,2,2,22,1 1,1,1,21,1 1 : det     Proof q.e.d.             =                                   =⋅ A A A CCCC CCCC CCCC aaaa aaaa aaaa aaaa AadjA nnninn ni ni nnninn iniiii ni ni det00 0det0 00det ,,,2,1 2,2,2,22,1 1,1,1,21,1 21 21 222221 111211                  ≠ = ==∑= ik ikA ACa ik n j jikj 0 det det, 1 , δsince Therefore multiplying by A-1 and dividing by det A, we obtain A Aadj A det 1 =− A-1 exists if and only if det A ≠ 0, i.e., the n rows/columns of Anxn are Linearly Independent ( ) nIAAadjA det=⋅ Return to Characteristic Polynomial Return to Cayley-Hamilton adj A is the adjugate of A [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
91 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 10 A-1 the Inverse of Matrix A with det A ≠ 0 is unique and given by:               ==− nnninn ni ni CCCC CCCC CCCC Aadjwhere A Aadj A ,,,2,1 2,2,2,22,1 1,1,1,21,1 1 : det     Proof (continue – 1)  BABBIAABIAA n I BbytionMultiplicaLeft n n =⇒==⇒= −−− 111 A-1 exists if and only if det A ≠ 0, i.e., the n rows/columns of Anxn are Linearly Independent Uniqueness Assume that exists a second Matrix B such that BA=In and q.e.d. adj A is the adjugate of A [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
92 SOLO Matrices Gabriel Cramer (1704-1752) Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution. The solution is expressed in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. Given n linear equations with n variables x1, x2,…,xn nnnnknknn nnkk nnkk bxaxaxaxa bxaxaxaxa bxaxaxaxa =+++++ =+++++ =+++++     2211 222222121 111212111 Cramer’s Rule states that the solution of this equation is nk aaaa aaaa aaaa abaa abaa abaa x nnnknn nk nk nnnnn n n k ,,2,1det/det 21 222221 111211 21 222221 111211          =                         = if the determinant that we divide by is not equal zero. Determinant of a Square Matrix – det A or |A| Cramer’s Rule11
93 SOLO Matrices Proof of Cramer's Rule To prove the Cramer’s Rule we use just two properties of Determinants: 1.adding one column to another does not change the value of the determinant 2.multiplying every element of one column by a factor will multiply the value of the determinant by the same factor In the following determinant let replace the b1,b2,…,bn by their equation ( ) ( ) ( )             +++++ +++++ +++++ =             nnnnnknknnnn nnnkk nnnkk nnnnn n n axaxaxaxaaa axaxaxaxaaa axaxaxaxaaa abaa abaa abaa         221121 2222221212221 1112121111211 21 222221 111211 detdet By subtracting from the k column the first multiplied by x1, the second column multiplied by x2, and so on until the last column multiplied by xn, ( the value of the determinant will not change by Rule 1 above), and it is found to be equal to             =             =             nnnknn nk nk k Rule nnknknn nkk nkk nnnnn n n aaaa aaaa aaaa x axaaa axaaa axaaa abaa abaa abaa             21 222221 111211 2 21 222221 111211 21 222221 111211 detdetdet q.e.d. Determinant of a Square Matrix – det A or |A| Cramer’s Rule11
SOLO Matrices Determinant of a Square Matrix – det A or |A| Therefore The Cramer’s Rule can be rewritten as nkbC A aaaa aaaa aaaa abaa abaa abaa x n j jjk nnnknn nk nk nnnnn n n k ,,2,1 det 1 det/det 1 , 21 222221 111211 21 222221 111211          ==                         = ∑= bAb A Aadj b b b CCC CCC CCC A x x x x nnnnn n n n 12 1 ,,2,1 2,2,22,1 1,1,21,1 2 1 detdet 1 : − =⋅=                           =             =      This result can be derived directly by using             =             == nn b b b b x x x xbxA  2 1 2 1 , Multiply from left by A-1  bAxAA nI 11 −− = [ ] A bAadj bAx det 1 ⋅ == − Proof of Cramer's Rule (continue – 1) Cramer’s Rule11
95 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof q.e.d. nn nnnknnnn nk nk aaa aaaa aa a a aaa aaaa A          2211 21 2221 11 2222 111211 00 000 det 000 0 detdet =             =             = 12 The Determinant of a Triangular Matrix is given by the product of the elements on the Main Diagonal Use Laplace’s Formula nn nnnkn nnnknnnnnknn aaa aaa a aa aaaa aa a a aaaa aa a             2211 3 33 2211 32 3332 22 11 21 2221 11 00 det 00 000 det 00 000 det ==           =             =             [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
96 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof 13 The Determinant of a Matrix Multiplication is equal to the Product of the Determinants ( ) BABA detdetdet ⋅= Start with the Multiplication of a Diagonal Matrix and any Matrix B.             =                         = Bnnn B B nnnn m n nn rd rd rd bbb bbb bbb d d d BD         222 111 21 22221 11211 22 11 00 00 00 In computing the Determinant use Property No. 4 ( ) ( ) BD r r r ddd rd rd rd BD Bn B B nn Bnnn B B detdetdetdetdet 2 1 2211 4 222 111 ⋅=             =             =    [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
97 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof (continue -1) 13 The Determinant of a Matrix Multiplication is equal to the Product of the Determinants ( ) BABA detdetdet ⋅= We have shown that by Invertible Elementary operations a Matrix A can be transformed to a Diagonal Matrix D. Each operation is to add to a given row one other row multiplied by a scalar (rj+α ri → rj ). According to Property (6) the value of the Determinant is unchanged by those operations. ( ) AAEDAED detdetdet ==⇒= Therefore by doing the same Elementary Operations on (A B) Matrix we have: ( ) ( )( ) ( ) BABDBDBAEBA ADDAE detdetdetdetdetdetdet detdet ⋅=⋅=== == ( ) BDBD detdetdet ⋅= q.e.d. Diagonalization of A [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
98 SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof mxmnxn mxmmxn nxmnxn BA BC A detdet 0 det ⋅=      14 Block Matrices Determinants q.e.d. ( ) ( )                     =              =      −− mmxn nxmn mxmmxn nxmnxn mxmmxn nxmnxn mmxn nxmn mxmmxn nxmnxn mxmmxn nxmnxn ICB I B I I A ICB I B A BC A 11 0 0 0 0 00 0 00 ( ) ( )                     =      − mmxn nxmn mxmmxn nxmnxn mxmmxn nxmnxn mxmmxn nxmnxn ICB I B I I A BC A 1 11 0 det 0 0 det 0 0 det 0 det ( ) ( ) ( ) A I A I A Laplace mxnm mnxnxn Laplace mxmmxn nxmnxn det 0 0 det1 0 0 det 11 1 ==        ⋅=      −− −  ( ) ( ) ( ) mxm LaplaceLaplace mxmnmx xmnn Laplace mxmmxn nxmnxn B B I B I det1 0 0 det1 0 0 det 1 11 ⋅==        ⋅=      − −−  ( ) 1 0 det 1 Triangular Matrix mmxn nxmn ICB I =        − mxmnxn mxmmxn nxmnxn BA BC A detdet 0 det =     
99 SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof [ ] [ ]    −⋅ −⋅ =      −− −− existsBifCBDAB existsAifDACBA BC DA mxnmxmnxmnxnmxm nxmnxnmxnmxmnxn mxmmxn nxmnxn 11 11 detdet detdet det 15 Block Matrices Determinants q.e.d.                −               −       =      − − − − − − existsBif ICB CBDA B DI existsAif DCAB DAI IC A BC DA m n n m mxmmxn nxmnxn 1 1 1 1 1 1 0 0 0 0 ( ) ( ) ( )    − − =                −               −       =      −− −− − − − − − − existsBifCBDAB existsAifDCABA existsBif ICB CBDA B DI existsAif DCAB DAI IC A BC DA m n n m mxmmxn nxmnxn 11 1112 1 1 1 1 1 1 detdet detdet 0 det 0 det 0 det 0 det det
100 SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof ( ) ( )nxmmxnmmxnnxmn ABIBAI +=+ detdet 16 Block Matrices Determinants q.e.d. ( ) ( ) ( ) ( ) ( ) ( )nxmmxnmxmmxnnxmnxn nxmnxnmxnmxmmxnmxmnxmnxn mxmmxn nxmnxn ABIBAI AIBIBIAI IB AI +=+= +=+=      − −− detdet detdetdet 1 13 1 13 Sylvester's Determinant Theorem James Joseph Sylvester (1814 – 1987)
101 SOLO Matrices Determinant of a Square Matrix – det A or |A| 17 Cauchy - Binet Formula Jacques Philippe Marie Binet (1786 –1856) Augustin-Louis Cauchy (1789 –1857) Let A be an m×n matrix and B an n×m matrix (m≤n). Write [n] for the set { 1, ..., n }, and for the set of m-combinations of [n] (i.e., subsets of size m; there are of them). For , write A[m],S for the m×m matrix whose columns are the columns of A at indices from S, and BS,[m] for the m×m matrix whose rows are the rows of B at indices from S. The Cauchy–Binet formula then states ( ) [ ]( ) [ ]( ) [ ] ∑      ∈ = m nS mSSm BAAB ,, detdetdet                         = nmnn m m mnmm n n nxmmxn bbb bbb bbb aaa aaa aaa BA         21 22221 11211 21 22221 11211 If m=n, and we recover ( ) BAAB detdetdet =1=      n n
102 It was Cauchy in 1812 who used 'determinant' in its modern sense. Cauchy's work is the most complete of the early works on determinants. He reproved the earlier results and gave new results of his own on minors and adjoints. In the 1812 paper the multiplication theorem for determinants is proved for the first time although, at the same meeting of the Institut de France, Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by Cauchy. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86
103 SOLO Matrices Determinant of a Square Matrix – det A or |A| Example       − − = 210 121 A           − − = 03 11 12 B 4 03 12 20 11 03 11 21 12 11 12 10 21 det 323311 −= −− + − − − + − − − = −−  BA 4 17 13 det 17 13 03 11 12 210 121 −=      − −       − − =           − −       − − =BA Using Cauchy - Binet Formula we obtain: By multiplying the matrices A and B and computing det (AB), we obtain: 3 2 23 2 3 ,3,2 = ⋅ =      == nm 17 Cauchy - Binet Formula
104 SOLO Matrices Determinant of a Square Matrix – det A or |A| ( ) ( ) 11 detdet −− = AA18 Proof q.e.d. nIAA =−1 use ( ) ( ) ( ) ( ) ( ) AAAAAAIn det/1detdetdetdetdet1 11 11 1 1 =⇒⋅=== −−− 19 ( ) AAT detdet = Proof ( ) ( ) UDLUDLAUDLA detdetdetdetdet 11 ⋅⋅==⇒= L and U are Triangular Matrices with 1 on the Main Diagonal, and D is diagonal.  nndddDUDLAUDLA  2211 11 detdetdetdetdet ==⋅⋅=⇒= ( ) AdddLDUALDUA nn TTTTTTTT detdetdetdetdet 2211 11 11 ==⋅⋅=⇒=  q.e.d.
105 SOLO Matrices Determinant of the Vandermonde Matrix ( ) ( )∏≤<≤ −−− −=                 = nji ij n n nn n n xx xxx xxx xxx xxxVnxn 1 11 2 1 1 22 2 2 1 21 221 111 det,,,det       Vandermonde Matrix is a nxn Matrix that has in its j row the entries x1 j-1 x2 j-1 … xn j-1 Determinant of a Square Matrix – det A or |A| 20 Proof: Using elementary operation, let multiply the (j-1) row by –x1 and add to row j, starting with j=n, then (n-1) until j=1                 = −−− →+−→+−→+−→+−→+−→+− −−−−−−−− 11 2 1 1 22 2 2 1 21 111 11112122111111212211 n n nn n n rrrxrrrxrrrxrrrxrrrxrrrx xxx xxx xxx EEEVEEE nnnnnnnxnnnnnnn       jj x E jjj rrxr ↑↑−                     − =→− − 1 1000 010 0010 0001 1 11       ←j-1 ←j ( ) ( )1 1 1 11 det,det 22 − − − −=      = nn nn nn xx xx xxV xWe have:
106 SOLO Matrices Determinant of the Vandermonde Matrix Vandermonde Matrix is a nxn Matrix that has in its j row the entries x1 j-1 x2 j-1 … xn j-1 Determinant of a Square Matrix – det A or |A| Proof (continue – 1): Using fact (13) that determinant of a product of Matrices is the product of their determinants                 −− −− −− =                 −−−−−−− →+−→+−→+− −−−− 2 1 12 21 1 2 1 2 21 2 2 112 11 2 1 1 22 2 2 1 21 0 0 0 111111 1111212211 n n n n nn nn n n n nn n n rrrxrrrxrrrx xxxxxx xxxxxx xxxx xxx xxx xxx EEE nnnnnn            ( ) ( ) ( )               −− −− −− =                 −− −− −− =                 −−−− −−−−−−− →+−→+−→+− −−−− 2 1 12 21 1 2 1 2 21 2 2 112 2 1 12 21 1 2 1 2 21 2 2 112 11 2 1 1 22 2 2 1 21 111 det 0 0 0 111 det 111 detdetdetdet 1111212211 n n n n nn nn n n n n n nn nn n n n nn n n rrrxrrrxrrrx xxxxxx xxxxxx xxxx xxxxxx xxxxxx xxxx xxx xxx xxx EEE nnnnnn                       
107 SOLO Matrices Determinant of the Vandermonde Matrix Vandermonde Matrix is a nxn Matrix that has in its j row the entries x1 j-1 x2 j-1 … xn j-1 Determinant of a Square Matrix – det A or |A| Proof (continue – 2): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nnxnn n n n n n n n n n nn n n n nn n n nnxn xxVxxxx xx xx xxxx xxxxxx xxxxxx xxxx xxx xxx xxx xxxV ,,det 11 detdet 111 det,,,det 211112 22 2 2 112 4 1 2 12 2 2 1122 112 11 2 1 1 22 2 2 1 21 21                 −− −−−− −−− −−=               −−=               −− −− −− =                 = We obtained a recursive relation between the nxn Vandermonde Matrix V (x1, x2, … , xn) and the (n-1)x(n-1) Matrix V (x2, … ,xn), and by continuing the procedure, and because det V2x2 (xn-1,xn)=(xn-xn-1), we obtain ( ) ( )∏≤<≤ −−− −=                 = nji ij n n nn n n xx xxx xxx xxx xxxVnxn 1 11 2 1 1 22 2 2 1 21 221 111 det,,,det       q.e.d. Use Property (4) that if the elements of a row/column of the Matrix A have a common factor λ than the Determinant of A is equal to the product of λ and the Determinant of the Matrix obtained by dividing the previous row/column by λ.
108 SOLO Matrices Eigenvalues and Eigenvectors of Square Matrices Anxn The relation represents a Linear Transformation of the vector to . 11 nxnxnnx xAy = 1nxx 1nxy For the Square Matrix Anxn a nonzero Vector is an Eigenvector if there is a Scalar λ (called the Eigenvalue) such that: 1nxv 11 nxnxnxn vvA λ= To find the Eigenvalues and Eigenvectors we see that ( ) 01 =− nxnnxn vIA λ This equation has a solution iff the Matrix (Anxn-λ In) is singular or01 ≠nxv ( ) 0det =− nnxn IA λ This equation may be used to find the Eigenvalues λ.
109 SOLO Matrices Eigenvalues and Eigenvectors of Square Matrices Anxn ( ) ( ) 01det 1 1 ' 21 22221 11211 =+++−=             − − − − n nnn RuleLeibniz nnnn n n cc aaa aaa aaa      λλ λ λ λ The equation that may be used to find the Eigenvalues λ can be written as: The polynomial: ( ) ( ) ( ) ( ) ( )nn nn ccp λλλλλλλλλ −−−=+++= −  21 1 1: is called the Characteristic Polynomial of the Square Matrix Anxn, and it has degree n and therefore n Eigenvalues λ1, λ2,…, λn. However the Characteristic Equations need not have distinct solutions, there may be less than n distinct eigenvalues. If the matrix has real entries, the coefficients of the characteristic polynomial are all real. However, the roots are not necessarily real; they may include complex numbers with a non-zero imaginary component. However, there is at least one complex number λ solving the characteristic equation, even if the entries of the matrix A are complex numbers to begin with. (This existence of such a solution is known as the Fundamental Theorem of Algebra.) For a complex eigenvalue, the corresponding eigenvectors also have complex components. By Abel’s Theorem (1824) there are no algebraic formulae for the roots of a general polynomial with n > 4, therefore we need an iterative algorithm to find the roots.
110 SOLO Matrices Eigenvalues and Eigenvectors of Square Matrices Anxn Theorem: The n Eigenvectors of a Square Matrix Anxn that has distinct Eigenvalues are Linearly Independent. Proof: Let assume that we have k (2 ≤ k ≤ n) Linearly Dependent Eigenvvectors. Then there exist k nonzero constants αi (i=1,…,k) such that: ivvv ikkii ∀≠=++++ 0011 αααα  kivvvA iiiinxn ,,1,0 =≠= λwhere: we have: ( ) ( ) ( ) 10 0 111 11111 ≠≠−=−=− =−=− iifvvvAvIA vvAvIA iiiinxninnxn nxnnnxn λλλλ λλ ( ) ( ) ( ) ( ) ( ) 0112122111 =−++−++−=++++− kkkiiikkiinnxn vvvvvvIA λλαλλαλλααααλ  In the same way multiplying the result by (Anxn – λ2 In) we obtain: ( ) ( ) ( )[ ] ( ) ( ) ( )( ) 012213233121222 =−−++−−=−++−− kkkkkkknnxn vvvvIA λλλλαλλλλαλλαλλαλ  Continuing the procedure until, at the end, we multiply by (Anxn – λ(k-1) In) to obtain: ( )  00 0 0 1 1 =⇒=       − ≠ ≠ − = ∏ kk k i ikk v αλλα    This contradicts the assumption that αk ≠ 0 therefore the k Eigenvectors are Linearly Independent.
111 SOLO Matrices Eigenvalues and Eigenvectors of Square Matrices Anxn Theorem: If the n Eigenvectors of a Square Matrix Anxn corresponding to the n Eigenvalues (not necessary distinct) are Linear Independent than we can write Proof:             =Λ=− n PAP λ λ λ     00 00 00 2 1 1 Using the n Eigenvectors of a Square Matrix Anxn we can write [ ] [ ] [ ]n n nn P n vvvvvvvvvA         21 2 1 221121 00 00 00             == λ λ λ λλλ or PPA Λ= Λ=− PAP 1 q.e.d. we say that the Square Matrix Anxn is Diagonalizable. Since the n Eigenvectors of Anxn are Linear Independent P is nonsingular and we have nvvv ,,, 21  Two Square Matrices A and B that are related by A=S-1 B S are called Similar Matrices Return to Matrix Decomposition
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Matrices i

  • 1. 1 Matrices I SOLO HERMELIN Updated: 30.03.11http://www.solohermelin.com
  • 2. 2 SOLO Matrices I Table of Content Introduction to Algebra Matrices Vectors and Vector Spaces Matrix Operations with Matrices Domain and Codomain of a Matrix A Transpose AT of a Matrix A Conjugate A* and Conjugate Transpose AH =(A* )T of a Matrix A Sum and Difference of Matrices A and B Multiplication of a Matrix by a Scalar Multiplication of a Matrix by a Matrix Kronecker Multiplication of a Matrix by a Matrix Partition of a Matrix Elementary Operations with a Matrix Rank of a Matrix Equivalence of Two Matrices
  • 3. 3 SOLO Matrices I Table of Content (continue – 1) Matrices Square Matrices Trace of a Square Matrix, Diagonal Square Matrix Identity Matrix, Null Matrix, Triangular Matrices Hessenberg Matrix Toeplitz Matrix, Hankel Matrix Householder Matrix Vandermonde Matrix Hermitian Matrix, Skew-Hermitian Matrix, Unitary Matrix Matrices & Determinants History L, U Factorization of a Square Matrix A by Elementary Operations Invertible Matrices Diagonalization of a Square Matrix A by Elementary Operations
  • 4. 4 SOLO Matrices I Table of Content (continue – 2) Matrices Square Matrices Determinant of a Square Matrix – det A or |A| Eigenvalues and Eigenvectors of Square Matrices Anxn Jordan Normal (Canonical) Form Cayley-Hamilton Theorem Matrix Decompositions Companion Matrix References
  • 5. 5 SOLO Algebra Set and Set Operations A collection of objects sharing a common property is called a Set. We use the notation { }PpropertyhasxxS := We write Sx ∈ S1 is a subset of S if every element of S1 is an element of S { }SxSxxSS ∈→∈∀=⊂ 11 : x is an element of S 1 2 { }elementsno=∅ Null (Empty) set { }2121 : SxorSxxSS ∈∈= Union of sets3 { }2121 : SxandSxxSS ∈∈= Intersection of sets4 { }2121 : SxandSxxSS ∉∈=− Difference of sets5 { }Ω=Ω∈∉= SSandxandSxxS : Complement of S relative to Ω 6 21 SS  1 S 2 S 21 SS  1 S 2 S 21 SS − 1 S 2 S Ω S S
  • 6. 6 SOLO Algebra Set and Set Operations A collection of objects sharing a common property is called a Set. We use the notation { }PpropertyhasxxS := We write Sx ∈ S1 is a subset of S if every element of S1 is an element of S { }SxSxxSS ∈→∈∀=⊂ 11 : x is an element of S 1 2 { }elementsno=∅ Null (Empty) set { }2121 : SxorSxxSS ∈∈= Union of sets3 { }2121 : SxandSxxSS ∈∈= Intersection of sets4 { }2121 : SxandSxxSS ∉∈=− Difference of sets5 { }Ω=Ω∈∉= SSandxandSxxS : Complement of S relative to Ω 6 21 SS  1 S 2 S 21 SS  1 S 2 S 21 SS − 1 S 2 S Ω S S
  • 7. 7 SOLO Algebra Group A nonempty set G is said to be a group if in G there is defined an operation * such that: GbaGba ∈∀∈ ,* Closure1 ( ) ( ) Gcbacbacba ∈∀= ,,**** Associativity2 3 GaaaeeatsGe ∈∀==∈∃ **.., Identity element 4 eabbatsGbGa ==∈∃∈∀ **..,, Inverse element b = a-1 Lemma1: A group G has exactly one identity element Proof: If e and f are both identity elements, then fe ffeef eeffe =⇒    == == ** ** Lemma2: Every element in G has exactly one inverse element Proof: If b and c are both inverse elements of x, then cxebx ** ==   cxbbxb ee **** = *b → → cebe ** = → cb =
  • 8. 8 SOLO Algebra Ring A Ring is a set R equipped with two binary operations +: R ×R→R (called addition), and •: R ×R→R (called multiplication), such that: (R,+) is an Abelian Group with identity element 0: ( ) ( )cbacba ++=++ aaa =+=+ 00 abba +=+ 0..,, =+−=−+∈−∃∈∀ aaaatsRaRa (R,.) is associative ( ) ( )cbacba ••=•• Multiplication distributes over addition: ( ) ( ) ( )cabacba •+•=+• ( ) ( ) ( )cbcacba •+•=•+ RbaRba ∈∀∈+ ,, Closure Associativity Identity element Inverse element Group Properties Abelian Group property
  • 9. 9 SOLO Algebra Field A Field is a Ring satisfying two additional conditions: (1) There also exists an identity e with respect to multiplication, i.e.: aaa =•=• 11 (2) All but the zero element have inverse with respect to multiplication 1..,,0& 111 =•=•∈=∃≠∈∀ −−− aaaatsRabaRa
  • 10. 10 Synthetic Geometry Euclid 300BC Algebras HistorySOLO Extensive Algebra Grassmann 1844 Binary Algebra Boole 1854 Complex Algebra Wessel, Gauss 1798 Spin Algebra Pauli, Dirac 1928 Syncopated Algebra Diophantes 250AD Quaternions Hamilton 1843 Tensor Calculus Ricci 1890 Vector Calculus Gibbs 1881 Clifford Algebra Clifford 1878 Differential Forms E. Cartan 1908 First Printing 1482 http://modelingnts.la.asu.edu/html/evolution.html Geometric Algebra and Calculus Hestenes 1966 Matrix Algebra Cayley 1854 Determinants Sylvester 1878 Analytic Geometry Descartes 1637 Table of Content
  • 11. 11 SOLO Matrices Definitions: Vectors and Vector Spaces Vector: A n-dimensional n-Vector is an ordered set of elements x1, x2,…,xn over a field F. One other way is to define it as Row Matrix or a Column Matrix [ ]n n xxxr x x x c   21 2 1 , =             = we have where T is the Transpose operation.crrc TT == & Scalar: A one-dimensional Vector with its element a real or a complex number. Null Vector: A n-dimensional Vector with all elements equal zero. Equality of two Vectors: niforyxyx ii ,1==⇔= [ ]000, 0 0 0   =             = rc oo
  • 12. 12 SOLO Matrices 12 VECTOR SPACE Given the complex numbers . A Vector Space V (Linear Affine Space) with elements over C if its elements satisfy the following conditions: I. Exists a operation of Addition with the following properties: Commutative (Abelian) Law for Addition1 Associative Law for Addition2 Exists a unique vector3 II. Exists a operation of Multiplication by a Scalar with the following properties: 4 Inverse 5 Associative Law for Multiplication6 Distributive Law for Multiplication7 Commutative Law for Multiplication8 We can write: ( ) ( ) ( ) ( ) ( ) 00101010 3 575 =⋅→==+=⋅+⋅=⋅+ xxxxxxxx ( ) yxyx βαα +=+ ( ) xxx βαβα +=+ ( ) ( )xx βαβα = xx =⋅1 0.. =+∈∃∈∀ yxtsVyVx xx =+ 0 0 ( ) ( )zyxzyx ++=++ xyyx +=+ Vzyx ∈,, C∈γβα ,,
  • 13. 13 SOLO Matrices Linear Dependence and Independence Vectors and Vector Spaces Vectors are said to be Linear Independent if:mvvv ,,, 21  00 212211 =====+++ mmm ifonlyandifvvv αααααα  Vectors are said to be Linear Dependent if :mvvv ,,, 21  0&02211 ≠=+++ imm somevvv αααα  k m ki i iik vv αα / 1           −= ∑ ≠ = If the vectors are Linear Dependent, the vectors whose coefficients αk ≠ 0 in can be obtained as a Linear Combination of other Vectors mvvv ,,, 21  kv 011 =++++ mmkk vvv ααα 
  • 14. 14 SOLO Matrices Linear Dependence and Independence Vectors and Vector Spaces Theorem If Vectors are said to be Linear Independent and vectors are Linear Dependent, than can be expressed as a Unique Linear Combination of . mvvv ,,, 21  121 ,,,, +mm vvvv  mvvv ,,, 21  1+mv Proof 0&0 1112211 ≠=++++ +++ mmmmm vvvv ααααα  since αm+1 = 0 implies mvvv ,,, 21  are Linear Dependent, and this is a contradiction. therefore: ( ) 122111 / ++ +++−= mmmm vvvv αααα  q.e.d. 121 ,,,, +mm vvvv  Linear Dependent implies that exists some (more than one) αi ≠ 0 s.t. To prove Uniqueness suppose that there are two expressions ( ) nivvvv ii tIndependenLinearvvm i iii m i ii m i iim m ,10 ,, 111 1 1 =∀=⇒=−⇒== ∑∑∑ === + γβγβγβ 
  • 15. 15 SOLO Matrices Basis of a Vector Space V Vectors and Vector Spaces A set of Vectors of a n-Vector Space is called a Basis of V if these n Vectors are Linearly Independent and every Vector can be Uniquely expressed as a Linear Combination of those Vectors: nvvv ,,, 21  y ∑= = n i iivy 1 α
  • 16. 16 SOLO Matrices Vectors and Vector Spaces Relation Between Two Bases of a Vector Space V If we have Two Bases of Vectors , we can writenn wwwandvvv ,,,,,, 2121                              =               ⇒        +++= +++= +++= n A nnnn n n nnnnnnn nn nn v v v w w w vvvw vvvw vvvw nxn              2 1 21 22221 11211 2 1 2211 22221212 12121111 ααα ααα ααα ααα ααα ααα In the same way                             =               ⇒        +++= +++= +++= n B nnnn n n nnnnnnn nn nn w w w v v v wwwv wwwv wwwv nxn              2 1 21 22221 11211 2 1 2211 22221212 12121111 βββ βββ βββ βββ βββ βββ Therefore               =               =               n nxnnxn n nxn n v v v AB w w w B v v v  2 1 2 1 2 1 Bnxn is called the Inverse of the Square Matrix Anxn and is written as Anxn -1 .               =               =               n nxnnxn n nxn n w w w BA v v v A w w w  2 1 2 1 2 1 nnxnnxn IBA = nnxnnxn IAB =
  • 17. 17 SOLO Inner Product If V is a complex Vector Space, for the Inner Product (a scalar) < , > between the elements (complex numbers) is defined by: Vzyx ∈∀ ,, * ,, >>=<< xyyx1 Commutative law ><+>>=<+< zxyxzyx ,,,2 Distributive law Cyxyx ∈∀><>=< λλλ ,,3 00,&0, =⇔=><≥>< xxxxx4 Using to we can show that:1 4 ( ) ( ) ( ) ><+><=><+><=>+<=>+< xyxyyxyxyyxxyy ,,,,,, 21 1 * 2 * 1 2 * 21 1 21 ( ) ( ) ><=><=><=>< yxxyxyyx ,,,, * 2 *** 2 λλλλ ( ) >=<>=<⇒><+><=>+>=<< xxxxxx ,000,0,0,00,0, 2 Matrices Vectors and Vector Spaces
  • 18. 18 SOLO Inner Product ( ) ** :, xyyxyx TT ==>< We can define the Inner Product in a Vector Space as Matrices therefore ( ) ∑= =+++>=<⇒             =             = n i iinn nn yxyxyxyxyx y y y y x x x x 1 ** 2 * 21 * 1 2 1 2 1 ,&   Outer Product ( ) [ ]               =             ==>< ** 2 * 1 * 2 * 22 * 12 * 1 * 21 * 11 ** 2 * 1 2 1 * : nnnn n n n n T yxyxyx yxyxyx yxyxyx yyy x x x yxyx       Vectors and Vector Spaces
  • 19. 19 SOLO (Identity) 00 =⇔= xx2 1 Vxx ∈∀≥ 0 (Non-negativity) xx λλ =4 Norm of a Vector .x Vyxyxyxyx ∈∀+≤+≤− ,3 (Triangle Inequalities) Matrices The Norm of a Vector is defined by the following relations: If V is an Inner Product space, than we can induce the norm: [ ] 2/1 , ><= xxx and We can see that 0, 2/1 1 2 2/1 1 *2/1 ≥      =      =>=< ∑∑ == n i i n i ii xxxxxx 1 0,100 2/1 1 2 =⇒=∀=⇒=      = ∑= xnixxx i n i i 2 Vectors and Vector Spaces
  • 20. 20 SOLO Inner Product yxyx ≤>< , Cauchy, Bunyakovsky, Schwarz Inequality known as Schwarz Inequality Let x, y be the elements of an Inner Product space V, than : 0,,,,, 2* ≥><+><+><+>>=<++< yyxyyxxxyxyx ααααα Assuming that (for which the equality holds) we choose: >< >< −= yy yx , , α we have: 0, , , , ,, , ,, , 2 2* ≥>< >< >< + >< ><>< − >< ><>< −>< yy yy yx yy xyyx yy yxyx xx which reduce to: 0 , , , , , , , 222 ≥ >< >< + >< >< − >< >< −>< yy yx yy yx yy yx xx or: ><≥⇔≥><−><>< yxyxyxyyxx ,0,,, 2 q.e.d. Augustin Louis Cauchy )1789-1857( Viktor Yakovlevich Bunyakovsky 1804 - 1889 Hermann Amandus Schwarz 1843 - 1921 MatricesVectors and Vector Spaces 0≠y
  • 21. 21 SOLO Inner Product Cauchy Inequality Let ai, bi (i = 1,…,n) be complex numbers, than :             ≤ ∑∑∑ === n i i n i i n i ii baba 1 2 1 2 2 1 Augustin Louis Cauchy )1789-1857( Viktor Yakovlevich Bunyakovsky 1804 - 1889 Hermann Amandus Schwarz 1843 - 1921 Buniakowsky-Schwarz Inequality ( ) ( ) ( )[ ] ( )[ ]∫∫∫ ≤ dttgdttfdttgtf 22 2 Buniakowsky, V., “Sur quelques inéqualités concernant Les intégrales ordinaires et les intégrales aux différences finite”, Mémoires de l’Acad. de St. Pétersbourg (VII),(1859) Schwarz, H.A., “Über ein die Flächen kleinstein Flächeninhalts betreffendes Problem der Variationsrechnung”, Acta Soc. Scient. Fen., 15, 315-362, (1885) Matrices Vectors and Vector Spaces
  • 22. 22 SOLO Inner Product [ ] 2/1 , ><= xxx Parallelogram law Given an Inner Product space V, than is a norm on V. Moreover for any x,y є X the parallelogram law 2222 22 yxyxyx +=−++ is valid. Proof q.e.d. x y yx + yx − 22 22 22,2,2 ,,,, ,,,, ,, yxyyxx yyxyyxxx yyxyyxxx yxyxyxyxyxyx +>=<+><= ><+><−><−><+ ><+><+><+>=< >−−<+>++=<−++ Matrices Vectors and Vector Spaces
  • 23. 23 SOLO Inner Product Let compute: From this we can see that ><+><= ><−><+><+><− ><+><+><+>=< >−−<−>++=<−−+ xyyx yyxyyxxx yyxyyxxx yxyxyxyxyxyx ,2,2 ,,,, ,,,, ,, 22 ><+><−= ><−><+><−><− ><+><+><−>=< ><−><+><+><− ><+><+><+>=< >−−<−>++=<−−+ xyiyxi yyxyiyxixx yyxyiyxixx yiyixyiyixxx yiyixyiyixxx yixyixyixyixyixyix ,2,2 ,,,, ,,,, ,,,, ,,,, ,, 22 ><=−−++−−+ yxyixiyixiyxyx ,4 2222 *2222 ,4,4 ><>=<=−++−−−+ yxxyyixiyixiyxyx MatricesVectors and Vector Spaces
  • 24. 24 SOLO Norm of a Vector . Matrices Let use the Norm definition to develop the following relations: yxyx yyxxyx yyyxxyxxyxyxyx ,Re2 ,, ,,,,, 22 22 2 ++= +++= +++=++=+ We obtain the Triangle Inequalities yxyxyxyxyx ,2,2 22222 ++≤+≤−+ ( ) ( ) yxyxyxyx ,Re,Im,Re, 22 ≥+=use the fact that: to obtain: use the Scwarz Inequality: ><≥ yxyx , yxyxyxyxyx 22 22222 ++≤+≤−+to obtain: or: ( ) ( )222 yxyxyx +≤+≤− ( ) ( )yxyxyx +≤+≤− Vectors and Vector Spaces x
  • 25. 25 SOLO Norm of a Vector . Matrices Other Definitions of Vector Norms ∑= = n i ixx 1 The following definitions satisfy Vector Norm Properties: 1 2 { }i i xx max= ( ) ( )[ ] ( )[ ] [ ] ∑∑= = ==== n i n j jiij TT xxqxQxxTTxxTxTx 1 1 *2/1* 2/1 ** 2/1 **3 Vectors and Vector Spaces x Return to Table of Content
  • 26. 26 SOLO Matrices Matrix A Matrix A over a field F is a rectangular array of elements in F. If A is over a field of real numbers, A is called a Real Matrix. If A is over a field of complex numbers, A is called a Complex Matrix. A n rows by m columns Matrix A, n x m Matrix, is defined as: [ ]                       ==             = s w o r n r r r ccc aaa aaa aaa A n columnsm m nmnn m m nxm         2 1 21 21 22221 11211 aij (i=1,n,j=1,m) are called the elements of A, and we use also the notation: { }ijaAnxm = Return to Table of Content
  • 27. 27 SOLO Matrices Definitions: Any complex matrix A with n rows (r1, r2,…,rn) and m columns (c1,c2,…,cm) [ ]m n nxm ccc r r r A ,,, 21 2 1   =               = can be considered as a linear function (or mapping or transformation) for a m-dimensional domain to a n-dimensional codomain. ( ) ( ){ }AcodomyAdomxxAyA nxmxnxm ∈⇒∈= 11;: In the same way its conjugate transpose: [ ]H n HH H m H H H mxn rrr c c c A ,,, 21 2 1   =               = is a linear function (or mapping or transformation) for a n-dimensional codomain to a m-dimensional domain. ( ) ( ){ }AcdomxAcodomyyAxA mxnx HH mxn ∈⇒∈= 111111 ;: Operations with Matrices
  • 28. 28 SOLO Matrices Domain and Codomain of a Matrix A The domain of A can be decomposed into orthogonal subspaces: ( ) ( ) ( )ANARAdom H ⊥ ⊕= ( )H AR ( )AN ( )H AN ( )AR xAy = 11 yAx H = ( )Adomxmx ∈1 11mx x ( )Acodomy nx ∈11 1nx yR (AH ) – is the row space of AH (dimension r) N (A) – is the null-space of A (x∈ N (A) ⇔ A x = 0) or the kernel of A (ker (A)) (dimension m-r) The codomain of A (domain of AH ) can be decomposed into orthogonal subspaces: ( ) ( ) ( )H ANARAcodom ⊥ ⊕= R (A) – is the column space of A (dimension r) N (AH ) – is the null-space of AH (dimension n-r) Operations with Matrices Return to Table of Content
  • 29. 29 SOLO Matrices Operations with Matrices The Transpose AT of a Matrix A is obtained by interchanging the rows with the columns. For             = nmnn m m aaa aaa aaa Anxm     21 22221 11211 Transpose AT of a Matrix A the transpose is ( )             == nmmm n n TT aaa aaa aaa AA mxnnxm     21 22212 12111 From the definition it is obvious that (AT )T = A Return to Table of Content
  • 30. 30 SOLO Matrices Operations with Matrices The Conjugate AT of a Matrix A is obtained by tacking the conjugate complex of each of the elements of A. { }* ** 2 * 1 * 2 * 22 * 21 * 1 * 12 * 11 * ij nmnn m m a aaa aaa aaa A nxm =               =     Conjugate A* of a Matrix A the transpose is ( )               == ** 2 * 1 * 2 * 22 * 12 * 1 * 21 * 11 * nmmm n n TH aaa aaa aaa AA nxmmxn     Conjugate Transpose AH =(A* )T of a Matrix A Return to Table of Content
  • 31. 31 SOLO Matrices Operations with Matrices The sum/difference of two matrices A and B of the same dimensions n x m is obtained by adding/subtracting the elements bij to/from elements aij. Sum and Difference of Matrices A and B of the same dimensions n x m { }ijij nmnmnnnn mm mm ba bababa bababa bababa BA nxmnxm ±=             ±±± ±±± ±±± =±     2211 2222222121 1112121111 Given the following transformations 1111 , mxnxmnxmxnxmnx xBzxAy == ( ) 11111 mxnxmnxmmxnxmmxnxmnxnx xBAxBxAzy ±=±=± Return to Table of Content
  • 32. 32 SOLO Matrices Operations with Matrices Multiplication of a Matrix by a Scalar The product of a Matrix by a Scalar is a Matrix in which each Element is multiplied by the Scalar. { }ij nmnn m m a aaa aaa aaa Anxm α ααα ααα ααα α =             =     21 22221 11211 Given the following operations 1111 , mxnxmnxmxnxmnx xAzxAy α== Return to Table of Content
  • 33. 33 SOLO Matrices Operations with Matrices Multiplication of a Matrix by a Matrix Consider the two consecutive transformations: nxp npnn p p mpmm p p nmnn m m C ccc ccc ccc bbb bbb bbb aaa aaa aaa BA mxpnxm =               =                           =             21 22221 11211 21 22221 11211 21 22221 11211 where                             ===             pmpmm p p pxmx m z z z bbb bbb bbb zBx x x x mxp       2 1 21 22221 11211 11 2 1 11 2 1 21 22221 11211 1 2 1 pxmxpnxmmxnxm zBA x x x aaa aaa aaa xAy y y y mnmnn m m nx n =                         ===                  
  • 34. 34 SOLO Matrices Operations with Matrices Multiplication of a Matrix by a Matrix (continue -1) The Multiplication of a Matrix by a Matrix is possible between Matrices in which the number of the columns in the first Matrix is equal to the number of rows in the second Matrix . nxp npnn p p mpmm p p nmnn m m C ccc ccc ccc bbb bbb bbb aaa aaa aaa BA mxpnxm =               =                           =             21 22221 11211 21 22221 11211 21 22221 11211 where ∑= = m j jkijik bac 1 :
  • 35. 35 SOLO Matrices Operations with Matrices Multiplication of a Matrix by a Matrix (continue - 2) CABBCA )()( =Matrix multiplication is associative: Transpose of Matrix Multiplication TTT ABAB =)( Matrix product is compatible with scalar multiplication: ( ) ( )BABAAB ααα ==)( Matrix multiplication is distributive over matrix addition: ( ) CBCACBACABACBA +=++=+ ,)( In general Matrix Multiplication is not Commutative ABAB ≠ Return to Table of Content
  • 36. 36 SOLO Matrices Operations with Matrices Kronecker Multiplication of a Matrix by a Matrix ( ) ( )pmxrnnmnn m m rprr p p nmnn m m BaBaBa BaBaBa BaBaBa bbb bbb bbb aaa aaa aaa BA rxpnxm ⋅⋅             =               ⊗             =⊗             21 22221 11211 21 22221 11211 21 22221 11211 :Leopold Kronecker (1823 –1891) ( ) ( ) ( ) ( ) ( ) ( ) ( )CBACBA BABABA CBCACBA CABACBA ⊗⊗=⊗⊗ ⊗=⊗=⊗ ⊗+⊗=⊗+ ⊗+⊗=+⊗ ααα Properties Return to Table of Content
  • 37. 37 SOLO Matrices Operations with Matrices Partition of a Matrix ( ) ( ) ( ) ( )           =                         = −−− − + +++++ + + pmxqnxpqn pmqxqxp nxm AA AA aaaa aaaa aaaa aaaa A nmnpnpn mqpqpqq qmqpqpq mpp 1221 1211 11 111111 11 111111                     = qpq p aa aa A qxp    1 111 11 : ( )           = + + − qmqp mp aa aa A pmqx    1 111 12 : ( )           = ++ − npn pqq aa aa A xpqn    1 111 21 : ( ) ( )           = + +++ −− nmnp mqpq aa aa A pmxqn    1 111 12 :
  • 38. 38 SOLO Matrices Operations with Matrices Partition of a Matrix (continue) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         ++ ++ =                     = −−−−−−−−−− −−−−−− −−− − −−− − srxpmpmxqnsrpxxpqnxspmpmxqnpxsxpqn srxpmpmqxsrpxqxpxspmpmqxpxsqxp srxpmxspm srpxpxs pmxqnxpqn pmqxqxp mxrnxm BABABABA BABABABA BB BB AA AA BA 2222122121221121 2212121121121111 2221 1211 2221 1211       Return to Table of Content
  • 39. 39 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix j iEE ji cr ↑ ←                 == 100 00 001      ααα AE irα jcEA α The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible) The reverse operation is to multiply the row/column elements by the scalar inverse. ( ) ( ) j iEE ji cr ↑ ←                 == 100 0/10 001 /1/1      ααα ( ) ( ) nrrrr IEEAAEE iiii =⇒= αααα /1/1 ( ) ( ) mcccc IEEAEEA jjjj =⇒= αααα /1/1 1. Multiple the elements of a row/column by a nonzero scalar The reverse operations are written as: ( ) ( ) ( )( ) 1 /1 1 /1 & −− == iiii rrrr EEEE αααα ( ) ( ) ( )( ) 1 /1 1 /1 & −− == jjjj cccc EEEE αααα
  • 40. 40 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix (continue – 1) The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible)                       =→+ 1000 010 0010 0001        α α jij rrrE ←i ←j nrrrrrr IEE jijjij =                       =                                             − =→+→− 1000 0100 0010 0001 1000 010 0010 0001 1000 010 0010 0001                      αα αα The reverse operation is to multiply each element of row i by the scalar (-α) and add to elements of row j AAEE jijjij rrrrrr =→+→− αα                         ++++ =→+ nnnjnin injnijjjiijiij inijiii nji rrr aaaa aaaaaaaa aaaa aaaa AE jij        1 11 1 11111 αααα α ←i ←j 2.a Multiply each element of row i by the scalar α and add to elements of row j                       − =→− 1000 010 0010 0001        α α jij rrrE ←i ←j
  • 41. 41 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix (continue – 2) The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible) ji E jij ccc ↑↑                       =→+ 1000 0100 010 0001        α α ←i ←j ncccccc IEE jijjij =                       =                                             − =→+→− 1000 0100 0010 0001 1000 0100 010 0001 1000 0100 010 0001                      αα αα The reverse operation is to multiply each element of column i by the scalar (-α) and add to elements of column j AEEA jijjij cccccc =→+→− αα                       − =→− 1000 0100 010 0001        α α jij cccE ←i ←j ji aaaaa aaaaa aaaaa aaaaa EA nnninjnin jnjijjjij iniiijiii niji ccc jij ↑↑                         + + + + =→+        α α α α α 1 1 1 111111 ←i ←j 2.b Multiply each element of column i by the scalar α and add to elements of column j
  • 42. 42 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix (continue – 3) The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible)                       =↔ 1000 0010 0100 0001        ji rrE ←i ←j nrrrr IEE jiij =                       =                                             =↔↔ 1000 0100 0010 0001 1000 0010 0100 0001 1000 0010 0100 0001                      The reverse operation is again interchange row j with row i AAEE jiij rrrr =↔↔                         =↔ nnnjnin inijiii jnjjjij nji rr aaaa aaaa aaaa aaaa AE ji        1 1 1 11111 ←i ←j ( ) jiij rrrr EE ↔ − ↔ = 1 3.a Interchange row i with row j
  • 43. 43 SOLO Matrices Operations with Matrices Elementary Operations with a Matrix (continue – 4) The Elementary Operations on rows/columns of a Matrix Anxm are reversible (invertible) ji E ji rr ↑↑                       =↔ 1000 0010 0100 0001        ncccc IEE ijji =                       =                                             =↔↔ 1000 0100 0010 0001 1000 0010 0100 0001 1000 0010 0100 0001                      The reverse operation is again interchange column j with column i AEEA ijji cccc =↔↔ ji aaaa aaaa aaaa aaaa EA nnninjn jnjijjj iniiiji nij cc ji ↑↑                         =↔        1 1 1 11111 ( ) jiij cccc EE ↔ − ↔ = 1 3.b Interchange column i with column j Return to Table of Content
  • 44. 44 SOLO Matrices Operations with Matrices Rank of a Matrix Given a Matrix Anxm we want, by using Elementary (reversible) Operations to reduce it to a Main Diagonal Unit Matrix and zeros in all other positions.             = nmnn m m aaa aaa aaa Anxm     21 22221 11211 Assume that a11 ≠ 0. If this is not the case interchange the first row/column (a Elementary operation) until this is satisfied. Divide the elements of the first row by a11. For i=2,n multiply the first row by (–ai1/a11) and add to i row (a Elementary operation) to obtain:                       −− −− = m n nm n n mm m a a a aa a a a a a a aa a a a a a a a AE nxm 1 11 1 12 11 1 2 1 11 21 212 11 21 22 11 1 11 12 1 0 0 1    
  • 45. 45 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 1) Repeat this procedure for second column (starting at the new a22), third column (starting at the new a33), and so on, as long as we ca obtain non-zero elements on the main diagonal, using the rows bellow. At the end we obtain: r a aa aaa AEEE nm mr mr rowrowrrow nxm ↑                       = 0000 0000 '100 ''10 '''1 22 1112 1_2__         ←r Define the multiplications of Elementary Operations as: 1_2__: rowrowrrow EEEP = Those Elementary Operations can be reversed in opposite order to obtain: ( ) ( ) ( ) 1 _ 1 2_ 1 1_ 1 : −−−− = rrowrowrow EEEP  nIPP =−1
  • 46. 46 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 2) Now use column operation starting with the first column in order to nullify all the elements above the Main Unit Diagonal: ( ) ( ) ( ) ( )         =                       = −−− − rmxrnxrrn rmrxr rcccrowrowrrow I EEEAEEE nxm 00 0 0000 0000 0100 0010 0001 _2_1_1_2__         Define the multiplications of Elementary Operations as: rccc EEEQ _2_1_: = Those Elementary Operations can be reversed in opposite order to obtain: ( ) ( ) ( ) 1 1_ 1 2_ 1 _ 1 : −−−− = ccrc EEEQ  mIQQ =−1
  • 47. 47 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 3) We obtained:  1111 00 0 00 0 −−−−       =⇒      = Q I PQQAPP I QPA r I I r m nxm n nxm  The maximum number of Linearly Independent Rows of A = r 11 00 0 −−       = Q I PA r nxm From the relation we can see that the maximum number of Linearly Independent Rows and the maximum number of Linearly Independent Columns of Matrix PAQ is r.       = 00 0rI QPAnxm       =               =      = −− −− −− − 0000 0 00 0 12 1 11 1 22 1 21 1 12 1 11 1 1 QQ QQ QQI Q I PA rr nxm Since the maximum number of Linearly Independent Rows of Matrix PA is also r. But the Elementary Operations P are not changing the number of Linearly Independent Rows of A, therefore: The maximum number of Linearly Independent Columns of A = r         =              =      = − − −− −− − 0 0 00 0 00 0 21 1 11 1 22 1 21 1 12 1 11 1 1 P PI PP PPI PQA rr nxm Since the maximum number of Linearly Independent Columns of Matrix A Q is also r. But the Elementary Operations Q are not changing the number of Linearly Independent Columns of A, therefore:
  • 48. 48 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 4) We obtained:       = 00 0rI QPAnxm The maximum number of Linearly Independent Rows of Anxm = The maximum number of Linearly Independent Columns of Anxm = r ≤ min (m,n) := Rank of Matrix Anxm 11 00 0 −−       = Q I PA r nxm ( ) ( ) ( )TrT mxn TT P I QAAnxm 11 00 0 −−       == Since in the Transpose of A we interchanged the columns with the rows of A: nxmmxn T ARankARank =
  • 49. 49 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 5) Proof ( ) ( ) mxpmxp mxp BRankBARank ARankBARank nxm nxmnxm ≤ ≤ Rank of A B: ( )mnrARank nxm ,min≤=Assume:       =               =      =⇒      = −− −− −− − 0000 0 00 0 00 0 12 1 11 1 22 1 21 1 12 1 11 1 1 QQ QQ QQI Q I PA I QPA rrr nxmnxm ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         =                 = −−− − −−− − −−− − −− rpxrnxrrn rprxrxr rpxrmxrrm rprxrxr rmxrnxrrn rmrxrxr mxp HH BB BBQQ BPAnxm 0000 1211 1221 121112 1 11 1 Therefore (P A B) has at most r nonzero rows: ARankrABRankPABRank rNonsingulaP =≤= Since ( ) ( ) BRankBRankABRankABRankABRankABAB TTTTTTT =≤==⇒= q.e.d.
  • 50. 50 SOLO Matrices Operations with Matrices Rank of a Matrix (continue – 6) ( ) ( ) nxnnxnnxn nxnnxnnxn BRankARankmBARank BRankARankBARank nxn nxn +≤+ +≤+ If A and B are Square nxn Matrices then: [3] K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967, p.104 ( ) ( )nxpmxnmxnnxpmxn BRankARankBARanknBRankARank nxp ,min≤≤−+ Sylvester’s Inequality: James Joseph Sylvester (1814 – 1887) [4] T. Kailath, “Linear Systems”, Prentice Hall, Inc., 1980, p.654 Return to Table of Content
  • 51. 51 SOLO Matrices Operations with Matrices Equivalence of Two Matrices Proof Two Matrices Anxm and Bnxm are said to be Equivalent, if and only if there exist a Nonsingular Matrix Pnxn and a Nonsingular Matrix Qmxm such that A=P B Q. This is the same to saying that A and B are Equivalent if and only if they have the same rank. Since A and B have the same rank r, we can write: T I SBH I GA rr       =      = 00 0 , 00 0 where G,H, S, T are square invertible matrices. q.e.d. QBPHTBSGATBS I QP r ==⇒=      −−−−  1111 00 0 P and Q are square invertible matrices since ( ) ( ) THHTQGSSGPHTQSGP 1111111111 ,:&: −−−−−−−−−− ====⇒== Return to Table of Content
  • 52. 52 SOLO Matrices Square Matrices In a Square Matrix Number of Rows = Number of Columns = n             = nnnn n n aaa aaa aaa Anxn     21 22221 11211 Trace of a Square Matrix ∑= == n i iiaAtrAoftrace nxnnxn 1 Diagonal Square Matrix { }ijij nn a a a a Dnxn δ=             =     00 00 00 22 11 Return to Table of Content
  • 53. 53 SOLO Matrices Square Matrices Identity Matrix Triangular Matrices { }ijnnxn II δ=             == 100 010 001     A Matrix whose elements below or above the main diagonal are all zero is called a Triangular Matrix             = nnnn aaa aa a Lnxn     21 2221 11 0 00 nxnnxnnxnnxnnxn AAIIA == Null Matrix { }0=nxn O nxnnxnnxnnxnnxn OOIIO == Upper Triangular Matrix Lower Triangular Matrix             = nn n n a aa aaa Unxn     00 0 222 11211 Return to Table of Content
  • 54. 54 SOLO Matrices Square Matrices Hessenberg Matrix An Upper Hessenberg Matrix has zero entries below the first subdiagonal: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                         = − −− −− −− −− nnnn nnn nnn nnn nnn H aa aaaa aaaaa aaaaaa aaaaaa U nxn 1 4142443 313233332 21222232211 11121131211 0000 00 0       An Lower Hessenberg Matrix has zero entries below the first superdiagonal: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                         = −−−−− −−−− nnnnnn nnnnnn nnnn H aaaaa aaaaa aaaa aaaa aaa aa L nxn        4321 141312111 42322212 34333231 232221 1211 0 0 00 000 A Hessenberg Matrix is an “almost” Triangular Matrix. Return to Table of Content
  • 55. 55 SOLO Matrices Square Matrices Toeplitz Matrix A Toeplitz Matrix or a “Diagonal-constant Matrix”, named after Otto Toeplitz, is a Matrix in which each descending Diagonal from left to right is constant. Otto Toeplitz (1881 – 1940)                    = − − −− − +−−− 0121 101 21 012 101 1210 aaaa aaa aa aaa aaa aaaa T n n nxn       Hankel Matrix A Hankel Matrix is closed related to a Toeplitz Matrix (a Hankel Matrix is an upside-down Toeplitz Matrix), named after Hermann Hankel, is a Matrix in which each uprising Diagonal from left to right is constant.                       = −+−+− +− −− −−− +−−− nnn n n nxn aaa a aa aaa aaaa H 2121 12 32 321 1210        Hermann Hankel (1839 – 1873) Return to Table of Content
  • 56. 56 SOLO Matrices Householder Matrix nˆ ( )xnn T  ˆˆ ( )xnn T  ˆˆ x  'x  O A We want to compute the reflection of over a plane defined by the normal ( )1ˆˆˆ =nnn T x  From the Figure we can see that: ( ) ( ) xHxnnIxnnxx TT  =−=−= ˆˆ2ˆˆ2' 1ˆˆˆˆ2: =−= nnnnIH TT We can see that H is symmetric: ( ) HnnInnIH TTTT =−=−= ˆˆ2ˆˆ2 In fact H is also a rotation of around OA so it must be orthogonal, i.e. HT H=H HT =I. x  ( ) ( )  InnnnnnInnInnIHHHH TTTTTT =+−=−−== ˆˆˆˆ4ˆˆ4ˆˆ2ˆˆ2 1 Alston Scott Householder 1904-1993 Square Matrices Return to Table of Content
  • 57. 57 SOLO Matrices Square Matrices Vandermonde Matrix ( )                 = −−− 11 2 1 1 22 2 2 1 21 21 111 ,,, n n nn n n n xxx xxx xxx xxxVnxn       Vandermonde Matrix is a nxn Matrix that has in its j row the entries x1 j-1 x2 j-1 … xn j-1 Alexandre-Théophile Vandermonde 1735 - 1796 Return to Table of Content
  • 58. 58 SOLO Hermitian = Symmetric if A has real components Hermitian Matrix: AH = A, Symmetric Matrix: AT = A Matrices Pease, “Methods of Matrix Algebra”, Mathematics in Science and Engineering Vol.16, Academic Press 1965 Definitions: Adjoint Operation (H): AH = (A*)T (* is complex conjugate and T is transpose of the matrix) Skew-Hermitian = Anti-Symmetric if A has real components. Skew-Hermitian: AH = -A, Anti-Symmetric Matrix: AT =-A Unitary Matrix: UH = U-1, Orthonormal Matix: OT = O-1 Unitary = Orthonormal if A has real components. Charles Hermite 1822 - 1901 Square Matrices Hermitian Matrix, Skew-Hermitian Matrix, Unitary Matrix Return to Table of Content
  • 59. 59 SOLO Matrices Square Matrices Singular, Non-singular and Inverse of a Non-singular Square Matrix Anxn We obtained:       = 00 0rI QPAnxn 11 00 0 −−       = Q I PA r nxn Singular Square Matrix Anxn: r < n Only r rows/columns of A are Linearly Independent Non-singular Square Matrix Anxn: r = n The n rows/columns of A are Linearly Independent For a Non-singular Matrix (r=n): n I n IQQQPPQAPQQPQIPA n nxn ===⇒== −−−−−−− 1111111  and:  n I n IPPPQQPPQAQPQIPA n nxn ===⇒== −−−−−−− 1111111 The Matrix (Q P) is the Inverse of the Non-singular Matrix A: PQAnxn = −1 This result explains the Gauss–Jordan elimination algorithm that can be used to determine whether a given square matrix is invertible and to find the inverse Return to Table of Content
  • 60. 60 SOLO Matrices Invertible Matrices Matrix Inversion • Gauss–Jordan elimination is an algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. • An alternative is the LU decomposition which generates an upper and a lower triangular matrices which are easier to invert. • For special purposes, it may be convenient to invert matrices by treating mxn- by-mxn matrices as m-by-m matrices of n-by-n matrices, and applying one or another formula recursively (other sized matrices can be padded out with dummy rows and columns). • For other purposes, a variant of Newton's method may be convenient (particularly when dealing with families of related matrices, so inverses of earlier matrices can be used to seed generating inverses of later matrices). Square Matrices
  • 61. 61 SOLO Matrices Invertible Matrices Square Matrices Gaussian elimination, which first appeared in the text Nine Chapters on the Mathematical Art written in 200 BC, was used by Gauss in his work which studied the orbit of the asteroid Pallas. Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix. Sketch of the orbits of Ceres and Pallas, by Gauss http://www.math.rutgers.edu/~cherlin/History/Papers1999/ weiss.html Gauss published his methods in 1809 as "Theoria motus corporum coelestium in sectionibus conicus solem ambientium," or, "Theory of the motion of heavenly bodies moving about the sun in conic sections."
  • 62. 62 SOLO Matrices Invertible Matrices Gauss-Jordan elimination In Linear Algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is variation of Gaussian elimination. Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards. Matrices containing zeros below each pivot are said to be in row echelon form. Gauss–Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form. Every matrix has a reduced row echelon form, and Gauss–Jordan elimination is guaranteed to find it. Carl Friedrich Gauss (1777–1855) Wilhelm Jordan ( 1842–1899) See example Square Matrices
  • 63. 63 SOLO Matrices Invertible Matrices Gauss-Jordan elimination If the original square matrix, A, is given by the following expression:           − −− − = 210 121 012 33xA Then, after augmenting A Matrix by the Identity Matrix, the following is obtained: [ ]           − −− − = 100210 010121 001012 IA Perform the following: 1. row1 + row2 →row1 equivalent with left multiplication by           =→+ 100 010 011 121 rrrE [ ]           − −− − =→+ 100210 010121 011111 121 IAE rrr Square Matrices
  • 64. 64 SOLO Matrices Invertible Matrices Gauss-Jordan elimination 2. row1 + row2 →row2 equivalent with left multiplication by           =→+ 100 011 001 221 rrrE [ ]           − −− − =→+ 100210 010121 011111 121 IAE rrr [ ]           − − − =→+→+ 100210 021230 011111 121221 IAEE rrrrrr 3. (1/3) row2 →row2 equivalent with left multiplication by           = → 100 03/10 001 22 3 1 rr E [ ]             − − − =→+→+ → 100210 0 3 2 3 1 3 2 10 011111 121221 22 3 1 IAEEE rrrrrr rr 1. row1 + row2 →row1 Square Matrices
  • 65. 65 SOLO Matrices Invertible Matrices Gauss-Jordan elimination 3. (1/3) row2 →row2           =→+ 110 010 001 332 rrrE [ ]             − − − =→+→+ → 100210 0 3 2 3 1 3 2 10 011111 121221 22 3 1 IAEEE rrrrrr rr 4. row2+row3 →row3 equivalent with left multiplication by [ ]                 − − =→+→+ → →+ 1 3 2 3 1 3 4 00 0 3 2 3 1 3 2 10 011111 121221 22 332 3 1 IAEEEE rrrrrr rr rrr 5. row1-row2 →row1 equivalent with left multiplication by           − =→− 100 010 011 121 rrrE [ ]                   − − =→+→+ → →+→− 1 3 2 3 1 3 4 00 0 3 2 3 1 3 2 10 0 3 1 3 2 3 1 01 121221 22 332121 3 1 IAEEEEE rrrrrr rr rrrrrr Square Matrices
  • 66. 66 SOLO Matrices Invertible Matrices Gauss-Jordan elimination 5. row1-row2 →row1               = → 4 3 00 010 001 33 4 3 rr E [ ]                   − − =→+→+ → →+→− 1 3 2 3 1 3 4 00 0 3 2 3 1 3 2 10 0 3 1 3 2 3 1 01 121221 22 332121 3 1 IAEEEEE rrrrrr rr rrrrrr 6. (4/3) row3 →row3 equivalent with left multiplication by [ ]                   − − =→+→+ → →+→− → 4 3 2 1 4 1 100 0 3 2 3 1 3 2 10 0 3 1 3 2 3 1 01 121221 22 332121 33 3 1 4 3 IAEEEEEE rrrrrr rr rrrrrr rr 7. (1/3) row3+row1 →row1 equivalent with left multiplication by               = →+ 100 010 3 1 01 113 3 1 rrr E [ ]                   −=→+→+ → →+→− →→+ 4 3 2 1 4 1 100 0 3 2 3 1 3 2 10 4 1 2 1 4 3 001 121221 22 332121 33113 3 1 4 3 3 1 IAEEEEEEE rrrrrr rr rrrrrr rrrrr Square Matrices
  • 67. 67 SOLO Matrices Invertible Matrices Gauss-Jordan elimination 7. (1/3) row3+row1 →row1             = →+ 100 3 2 10 001 223 3 2 rrr E [ ]                   −=→+→+ → →+→− →→+ 4 3 2 1 4 1 100 0 3 2 3 1 3 2 10 4 1 2 1 4 3 001 121221 22 332121 33113 3 1 4 3 3 1 IAEEEEEEE rrrrrr rr rrrrrr rrrrr 8. (2/3) row3+row2 →row2 equivalent with left multiplication by [ ] [ ]BIIAEEEEEEEE B rrrrrr rr rrrrrr rrrrrrrr =                   =→+→+ → →+→− →→+→+ 4 3 2 1 4 1 100 2 1 1 2 1 010 4 1 2 1 4 3 001 121221 22 332121 33113223 3 1 4 3 3 1 3 2    We found [ ] [ ] 1− =⇒=⇒= ABIABBIIAB 1 3 1 4 3 3 1 3 2 4 3 2 1 4 1 2 1 1 2 1 4 1 2 1 4 3 : 121221 22 332121 33113223 − →+→+ → →+→− →→+→+ =                   == AEEEEEEEEB rrrrrr rr rrrrrr rrrrrrrr [ ] [ ]11 −− =− AIIAAinationlimeJordanGaussTherefore Square Matrices Return to Table of Content
  • 68. 68 The first to use the term 'matrix' was Sylvester in 1850. Sylvester defined a matrix to be an oblong arrangement of terms and saw it as something which led to various determinants from square arrays contained within it. After leaving America and returning to England in 1851, Sylvester became a lawyer and met Cayley, a fellow lawyer who shared his interest in mathematics. Cayley quickly saw the significance of the matrix concept and by 1853 Cayley had published a note giving, for the first time, the inverse of a matrix. Arthur Cayley 1821 - 1895 Cayley in 1858 published “Memoir on the Theory of Matrices” which is remarkable for containing the first abstract definition of a matrix. He shows that the coefficient arrays studied earlier for quadratic forms and for linear transformations are special cases of his general concept. Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses. He gave an explicit construction of the inverse of a matrix in terms of the determinant of the matrix. Cayley also proved that, in the case of 2 2 matrices, that a matrix satisfies its own characteristic equation. James Joseph Sylvester 1814 - 1897 http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86 Return to Table of Content
  • 69. 69 SOLO Matrices Square Matrices L, U Factorization of a Square Matrix A by Elementary Operations Given a Square Matrix Number of Rows = Number of Columns = n             = nnnn n n aaa aaa aaa Anxn     21 22221 11211 Consider the following Simple Operations on the rows/columns of A to obtain a U Triangular Matrix (all elements bellow the Main Diagonal are 0) : 1. Multiple the elements of a row/column by a nonzero scalar 2. Multiply each element of row i by the scalar α and add to elements of row j j iEE ji cr ↑ ←                 == 100 00 001      αααAE irα jcEA α                       =↔+ 1000 010 0010 0001        α α jij rrrE AE jij rrr →+α L,U factorization was proposed by Heinz Rutishauser in 1955.
  • 70. 70 SOLO Matrices Square Matrices L, U Factorization of a Matrix A by Elementary Operations Given a Square Matrix Number of Rows = Number of Columns = n for example:           − −− − = 210 121 012 33xA Consider the following Simple Operations on the rows/columns of A to obtain a U1 Triangular Matrix (all elements bellow the Main Diagonal are 0) :           = →+ 13/20 010 001 332 3 2 rrr E             − − − = →+ 210 1 2 3 0 012 221 2 1 AE rrr           = →+ 100 012/1 001 221 2 1 rrr E 1 2 1 3 2 3 4 00 1 2 3 0 012 221332 UAEE rrrrrr =                 − − = →+→+ 1. (1/2) row1+row2 →row1 equivalent with left multiplication by 2. (2/3) row2 + row3 →row3 equivalent with left multiplication by
  • 71. 71 SOLO Matrices Square Matrices L, U Factorization of a Matrix A by Elementary Operations           =                     = →→+ 13/23/1 012/1 001 100 012/1 001 13/20 010 001 11221 2 1 2 1 rrrrr EE 1 2 1 3 2 3 4 00 1 2 3 0 012 221332 UAEE rrrrrr =                 − − = →+→+ we found: To Undo the Simple Operations and to obtain again A, let perform:           − = →+ 13/20 010 001 332 3 2 rrr E we can see that:           =                     − = →+→+ 100 010 001 13/20 010 001 13/20 010 001 332332 3 2 3 2 rrrrrr EE 332 3 2 rrr E →+− is the Inverse Operation to and we write 1 3 2 3 2 332332 − →+→+−         = rrrrrr EE 332 3 2 rrr E →+           −= →+− 100 012/1 001 221 2 1 rrr E           =                     −= →+→+− 100 010 001 100 012/1 001 100 012/1 001 221221 2 1 2 1 rrrrrr EE 221 2 1 rrr E →+− is the Inverse Operation to and we write 1 2 1 2 1 221221 − →+→+−         = rrrrrr EE 221 2 1 rrr E →+ 1. (-2/3) row2 + row3 →row3 equivalent with left multiplication by 2. (-1/2) row1+row2 →row1 equivalent with left multiplication by
  • 72. 72 SOLO Matrices Square Matrices L, U Factorization of a Matrix A by Elementary Operations LEE rrrrrr =           − −=           −          −= →+−→+− 13/20 012/1 001 13/20 010 001 100 012/1 001 221332 2 1 3 2 AUEEAEEEE rrrrrrrrrrrrrrrrrr =        =                →+−→+−→+→+→+−→+− 1 3 2 2 1 2 1 3 2 3 2 2 1 332221221332332221 we found: Therefore we obtained an L U factorization of the Square Matrix A: AUL =           − −− − =                 − −           − −= 210 121 012 3 4 00 1 2 3 0 012 13/20 012/1 001 1 We can have 1 on the diagonal of U Matrix, by introducing the Diagonal Matrix D: UDLA =           − −                     − −=           − −− − = 100 2/310 02/11 4/300 03/20 002/1 13/20 012/1 001 210 121 012 Return to Table of Content
  • 73. 73 SOLO Matrices Square Matrices Diagonalization of a Square Matrix A by Elementary Operations we found: 1 2 1 3 2 3 4 00 1 2 3 0 012 221332 UAEE rrrrrr =                 − − = →+→+ 1. (3/2) row2 + row1 →row1 equivalent with left multiplication by 2. (4/3) row3 + row2 →row2 and (9/8) row3 + row1 →row1 equivalent with left multiplication by               = →+ 100 010 0 2 3 1 112 2 3 rrr E                   − − = →+→+→+ 3 4 00 1 2 3 0 2 3 02 221332112 2 1 3 2 2 3 AEEE rrrrrrrrr               = →+→+ 100 010 8 9 3 4 1 223113 3 4 8 9 rrrrrr EEDAEAEEEEE rrrrrrrrrrrrrrr =                 == →+→+→+→+→+ 3 4 00 0 2 3 0 002 221332112223113 2 1 3 2 2 3 3 4 8 9 Return to Table of Content
  • 74. 74 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             = To each Matrix A we associate a scalar called Determinant; i.e. det A or |A| defined by the following 4 properties: 1 The Determinant of the Identity Matrix In is 1. If the Matrix A has two identical rows/columns the Determinant of A is zero.2 0det 1 =                       nr r r r    α α [ ] 0det 1 =ncccc  αα ←i row ↑ i column 1 1000 0100 0010 0001 detdet =                     =       nI
  • 75. 75 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| To each Matrix A we associate a scalar called Determinant; i.e. det A or |A| defined by the following 4 properties: 3 If each element of a row/column of the Matrix A is the sum of two terms, the Determinant of A is the sum of the two Determinants formed by the separation of the terms                 +                 =                 + n k n k n kk r r r r r r r rr r       'detdet'det 111 [ ] [ ] [ ]nknknkk cccccccccc  'detdet'det 111 +=+ [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 76. 76 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 4 If the elements of a row/column of the Matrix A have a common factor λ than the Determinant of A is equal to the product of λ and the Determinant of the Matrix obtained by dividing the previous row/column by λ.                 =                 nknn knkk n nknn knkk n aaa aaa aaa aaa aaa aaa           21 21 11211 21 21 11211 detdet λλλλ [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 77. 77 SOLO Matrices & Determinants History http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86 The idea of a determinant appeared in Japan and Europe at almost exactly the same time although Seki in Japan certainly published first. In 1683 Seki wrote “Method of solving the dissimulated problems “ which contains matrix methods written as tables. Without having any word which corresponds to 'determinant' Seki still introduced determinants and gave general methods for calculating them based on examples. Using his 'determinants' Seki was able to find determinants of 2x2, 3x3, 4x4 and 5x5 matrices and applied them to solving equations but not systems of linear equations. Takakazu Shinsuke Seki 1642 - 1708Rather remarkably the first appearance of a determinant in Europe appeared in exactly the same year 1683. In that year Leibniz wrote to de l'Hôpital. He explained that the system of equations 10 + 11x + 12y = 0 20 + 21x + 22y = 0 30 + 31x + 32y = 0 had a solution because 302112322011312210312012302211322110 ⋅⋅+⋅⋅+⋅⋅=⋅⋅+⋅⋅+⋅⋅ which is exactly the condition that the coefficient matrix has determinant 0. Gottfried Wilhelm von Leibniz 1646 - 1716
  • 78. 78 SOLO Matrices & Determinants History http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86 Leibniz used the word 'resultant' for certain combinatorial sums of terms of a determinant. He proved various results on resultants including what is essentially Cramer's rule. He also knew that a determinant could be expanded using any column - what is now called the Laplace expansion. As well as studying coefficient systems of equations which led him to determinants, Leibniz also studied coefficient systems of quadratic forms which led naturally towards matrix theory. Gottfried Wilhelm von Leibniz 1646 - 1716 Gabriel Cramer (1704-1752) In the 1730's Maclaurin wrote Treatise of algebra although it was not published until 1748, two years after his death. It contains the first published results on determinants proving Cramer's rule for 2x2 and 3x3 systems and indicating how the 4x4 case would work. Cramer gave the general rule for n n systems in a paper Introduction to the analysis of algebraic curves (1750). It arose out of a desire to find the equation of a plane curve passing through a number of given points. Cramer does go on to explain precisely how one calculates these terms as products of certain coefficients in the equations and how one determines the sign. He also says how the n numerators of the fractions can be found by replacing certain coefficients in this calculation by constant terms of the system. Colin Maclaurin 1698 - 1746
  • 79. 79 An axiomatic definition of a determinant was used by Weierstrass in his lectures and, after his death, it was published in 1903 in the note ‘On Determinant Theory‘. In the same year Kronecker's lectures on determinants were also published, again after his death. With these two publications the modern theory of determinants was in place but matrix theory took slightly longer to become a fully accepted theory. Karl Theodor Wilhelm Weierstrass 1815 - 1897 Leopold Kronecker 1823 - 1891 Determinant Weirstrass Definition of Determinant of a nxn Matrix A: (1)det (A) is linear in the rows of A (2) Interchanging two rows change the sign of det (A) (3) det (In) = 1 For each positive integer n, there is exactly one function with these three properties. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86 http://www.sandgquinn.org/stonehill/MA251/notes/Weierstrass.pdf
  • 80. 80 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 5 If in a Matrix Determinant we interchange two rows/columns the sign of the Determinant will change. Proof [ ]nji cccc 1detgiven ( ) [ ] ( ) [ ] ( ) [ ]nji by nii njiji cccccccc cccccc       1 20 1 3 1 2 detdet det0 += ++= [ ] [ ] ( )     20 11 detdet by njjnij cccccccc ++ therefore [ ] [ ]nijnji cccccccc  11 detdet −= q.e.d. [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 81. 81 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 6 The Matrix Determinant is unchanged if we add to a row/column any linear combination of the other rows/columns. Proof [ ]nji cccc 1detgiven q.e.d. [ ] [ ] ( ) [ ]ni ij j by njjj nin ij j jji ccccccc ccccccc       1 20 1 11 detdet detdet =+ =           + ∑ ∑ ≠ ≠ λ λ [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 82. 82 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 7 If a row/column is a Linear Combination of other rows/columns the Determinant is zero. Proof q.e.d. [ ] ( ) 0detdet 20 11 ==           ∑∑ ≠≠ ij j by njjjn ij j jj ccccccc     λλ [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 83. 83 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 8 Leibniz formula for determinants ( ) ( )n n i n iii i n iii i nikii L nnnknn nk nk iiinPermutatioL aaa aaaa aaaa aaaa A kk k nn n nk ,,, 1detdet 21 1 21 222221 111211 1 11 11 1         = −=             = ∑ ∑ ∑ − −≠ ≠ The meaning of this equation is that in the product there are no two elements of the same row or the same column, and the sign of the product is a function of the position of each element in the Matrix. The sign of each element, in the product, is given by { } ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               −−−−− −−−+− −−+−+ =−= +++++ ++ ++ + nnknnnn nk nk ji ijasign 11111 11 11 1 321 22 11     Gottfried Wilhelm Leibniz (1646 – 1716) [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 84. 84 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof 8 ( ) ( )n n i n iii i n iii i nikii L nnnknn nk nk iiinPermutatioL aaa aaaa aaaa aaaa A kk k nn n nk ,,, 1detdet 21 1 21 222221 111211 1 11 11 1         = −=             = ∑ ∑ ∑ − −≠ ≠ From Properties (3) and (4) of the Determinant: ( ) ( ) ( ) ( ) [ ]010:detdetdetdet 1 321 4,3 1 2 1 4,3 21 222221 111211 1 2 2 1 21 1 1 1        =                   =               =             = ∑ ∑∑ == i n i n i n i i ii n i n i i nnnknn nk nk ewhere r r e e aa r r e a aaaa aaaa aaaa A ↑ i column ↑ 1st row coeff 2nd row Coeff ↓ From Properties (2) if two rows are identical the determinant is zero, therefore, in the summation of i2 we can delete the case i2=i1. [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 85. 85 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof (continue 1) 8 ( ) ( )n n i n iii i n iii i nikii L nnnknn nk nk iiinPermutatioL aaa aaaa aaaa aaaa A kk k nn n nk ,,, 1detdet 21 1 21 222221 111211 1 11 11 1         = −=             = ∑ ∑ ∑ − −≠ ≠ From Properties (2),(3) and (4) of the Determinant: ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ∑∑ ∑ ∑ ∑∑ ≠ ≠ == −               = =                   =               =             = n i n ii i n iiii i i i i niii i n i n i n i i ii n i n i i nnnknn nk nk nn n n n e e e aaa ewhere r r e e aa r r e a aaaa aaaa aaaa A 1 12 2 121 2 1 21 1 2 2 1 21 1 1 1 ,,, 21 4,3,2 1 321 4,3 1 2 1 4,3 21 222221 111211 det 010:detdetdetdet           ↑ i column [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 86. 86 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof (continue 1) q.e.d. 8 ( ) ( )n n i n iii i n iii i nikii L nnnknn nk nk iiinPermutatioL aaa aaaa aaaa aaaa A kk k nn n nk ,,, 1detdet 21 1 21 222221 111211 1 11 11 1         = −=             = ∑ ∑ ∑ − −≠ ≠ Let interchange the position of the rows to obtain a Unit Matrix, where, according with Property (5), each interchange will cause a change in determinant sign. We also use Property (1) that the determinant of the Unit Matrix is 1: ( )∑ ∑ ∑ − −≠ ≠ −=             = n i n iii i n iii i nikii L nnnknn nk nk kk k nn n nk aaa aaaa aaaa aaaa A 1 11 11 1 ,, ,, 1 21 222221 111211 1detdet        ( ) ( ) ( )L n L i i i e e e e e e n 1det1det 1 2 1 2 1 −=             −=                where L is the Number of Permutations necessary to go from (i1,i2,…,in) to (1,2,…,n) [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 87. 87 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Using the 4 properties that define the Determinant of a Square Matrix more properties can be derived 9 A Determinant can be expanded along a row or column using Laplace's Formula: ( )∑∑ = + = −== n k ki ki ik n k kiik MaCaA 1 , 1 , 1det where the Ci,k represents the i,k element of the matrix cofactors, i.e. Ci,k is ( − 1)i + k times the minor Mi,k, which is the determinant of the matrix that results from A by removing the i-th row and the k-th column, and n is the length of the matrix. Pierre-Simon, marquis de Laplace 1749 - 1827 ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )                         = +− ++++−++ +− −+−−−−− +− nnknnkknn nikikikii inkiikkii nikikikii nkkk ki aaaaa aaaaa aaaaa aaaaa aaaaa M        111 11111111 111 11111111 11111111 , det [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 88. 88 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 9 Laplace's Formula: ( ) ∑∑∑ == + = =−== n j ijji n j ji ji ij n j jiij CaMaCaA 1 , 1 , 1 , 1det Proof ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑= +− +− +−                     =             = n k nnknkknn nkkk nkkk ik nnnknn nk nk aaaaa aaaaa aaaaa a aaaa aaaa aaaa A 1 1111 21121221 11111111 4,3 21 222221 111211 00100 detdetdet           From Properties (3) and (4) of the Determinant, using Row summation: From Properties (3) and (5) of the Determinant: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                 +               −++               ⋅=                     +− +− +− + +− +− +− +− +− +− nnkknn nkk nkk ki nnknkknn nkkk nkkk nnknkknn nkkk nkkk aaaa aaaa aaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa 1111 2111221 1111111 1112 21121222 11111112 1111 21121221 11111111 det1det0 00100 det ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ki ki nnkknn nkk nkk ki nnknkknn nkkk nkkk M aaaa aaaa aaaa aaaaa aaaaa aaaaa , 1111 2111221 1111111 1111 21121221 111111111 1det1det0 + +− +− +− + +− +− −+− −=               −=               ⋅+          [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 89. 89 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 9 Laplace's Formula: ( ) ∑∑∑ == + = =−== n j ijji n j ji ji ij n j jiij CaMaCaA 1 , 1 , 1 , 1det ( )∑∑ = + = −== n j ji ji ij n j jiij MaCaA 1 , 1 , 1det Proof (continue 1) Therefore the minor Mi,k, which is the determinant of the matrix that results from A by removing the i-th row and the k-th column. We obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ki ki nnknkknn nkkk nkkk ki nnknkknn nkkk nkkk ki M aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa C , 1111 21121221 11111111 1111 21121221 11111111 , 1: 00100 det1 00100 det: + +− +− +− + +− +− +− −=                     −=                     =             q.e.d. In the same way we can use Column summation to obtain ( )∑∑ = + = −== n j ij ji ji n j ijji MaCaA 1 , 1 , 1det [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 90. 90 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 10 A-1 the Inverse of Matrix A with det A ≠ 0 is unique and given by:               ==− nnninn ni ni CCCC CCCC CCCC Aadjwhere A Aadj A ,,,2,1 2,2,2,22,1 1,1,1,21,1 1 : det     Proof q.e.d.             =                                   =⋅ A A A CCCC CCCC CCCC aaaa aaaa aaaa aaaa AadjA nnninn ni ni nnninn iniiii ni ni det00 0det0 00det ,,,2,1 2,2,2,22,1 1,1,1,21,1 21 21 222221 111211                  ≠ = ==∑= ik ikA ACa ik n j jikj 0 det det, 1 , δsince Therefore multiplying by A-1 and dividing by det A, we obtain A Aadj A det 1 =− A-1 exists if and only if det A ≠ 0, i.e., the n rows/columns of Anxn are Linearly Independent ( ) nIAAadjA det=⋅ Return to Characteristic Polynomial Return to Cayley-Hamilton adj A is the adjugate of A [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 91. 91 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| 10 A-1 the Inverse of Matrix A with det A ≠ 0 is unique and given by:               ==− nnninn ni ni CCCC CCCC CCCC Aadjwhere A Aadj A ,,,2,1 2,2,2,22,1 1,1,1,21,1 1 : det     Proof (continue – 1)  BABBIAABIAA n I BbytionMultiplicaLeft n n =⇒==⇒= −−− 111 A-1 exists if and only if det A ≠ 0, i.e., the n rows/columns of Anxn are Linearly Independent Uniqueness Assume that exists a second Matrix B such that BA=In and q.e.d. adj A is the adjugate of A [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 92. 92 SOLO Matrices Gabriel Cramer (1704-1752) Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution. The solution is expressed in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. Given n linear equations with n variables x1, x2,…,xn nnnnknknn nnkk nnkk bxaxaxaxa bxaxaxaxa bxaxaxaxa =+++++ =+++++ =+++++     2211 222222121 111212111 Cramer’s Rule states that the solution of this equation is nk aaaa aaaa aaaa abaa abaa abaa x nnnknn nk nk nnnnn n n k ,,2,1det/det 21 222221 111211 21 222221 111211          =                         = if the determinant that we divide by is not equal zero. Determinant of a Square Matrix – det A or |A| Cramer’s Rule11
  • 93. 93 SOLO Matrices Proof of Cramer's Rule To prove the Cramer’s Rule we use just two properties of Determinants: 1.adding one column to another does not change the value of the determinant 2.multiplying every element of one column by a factor will multiply the value of the determinant by the same factor In the following determinant let replace the b1,b2,…,bn by their equation ( ) ( ) ( )             +++++ +++++ +++++ =             nnnnnknknnnn nnnkk nnnkk nnnnn n n axaxaxaxaaa axaxaxaxaaa axaxaxaxaaa abaa abaa abaa         221121 2222221212221 1112121111211 21 222221 111211 detdet By subtracting from the k column the first multiplied by x1, the second column multiplied by x2, and so on until the last column multiplied by xn, ( the value of the determinant will not change by Rule 1 above), and it is found to be equal to             =             =             nnnknn nk nk k Rule nnknknn nkk nkk nnnnn n n aaaa aaaa aaaa x axaaa axaaa axaaa abaa abaa abaa             21 222221 111211 2 21 222221 111211 21 222221 111211 detdetdet q.e.d. Determinant of a Square Matrix – det A or |A| Cramer’s Rule11
  • 94. SOLO Matrices Determinant of a Square Matrix – det A or |A| Therefore The Cramer’s Rule can be rewritten as nkbC A aaaa aaaa aaaa abaa abaa abaa x n j jjk nnnknn nk nk nnnnn n n k ,,2,1 det 1 det/det 1 , 21 222221 111211 21 222221 111211          ==                         = ∑= bAb A Aadj b b b CCC CCC CCC A x x x x nnnnn n n n 12 1 ,,2,1 2,2,22,1 1,1,21,1 2 1 detdet 1 : − =⋅=                           =             =      This result can be derived directly by using             =             == nn b b b b x x x xbxA  2 1 2 1 , Multiply from left by A-1  bAxAA nI 11 −− = [ ] A bAadj bAx det 1 ⋅ == − Proof of Cramer's Rule (continue – 1) Cramer’s Rule11
  • 95. 95 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof q.e.d. nn nnnknnnn nk nk aaa aaaa aa a a aaa aaaa A          2211 21 2221 11 2222 111211 00 000 det 000 0 detdet =             =             = 12 The Determinant of a Triangular Matrix is given by the product of the elements on the Main Diagonal Use Laplace’s Formula nn nnnkn nnnknnnnnknn aaa aaa a aa aaaa aa a a aaaa aa a             2211 3 33 2211 32 3332 22 11 21 2221 11 00 det 00 000 det 00 000 det ==           =             =             [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 96. 96 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof 13 The Determinant of a Matrix Multiplication is equal to the Product of the Determinants ( ) BABA detdetdet ⋅= Start with the Multiplication of a Diagonal Matrix and any Matrix B.             =                         = Bnnn B B nnnn m n nn rd rd rd bbb bbb bbb d d d BD         222 111 21 22221 11211 22 11 00 00 00 In computing the Determinant use Property No. 4 ( ) ( ) BD r r r ddd rd rd rd BD Bn B B nn Bnnn B B detdetdetdetdet 2 1 2211 4 222 111 ⋅=             =             =    [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 97. 97 Let A be any square n by n matrix over a field F SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof (continue -1) 13 The Determinant of a Matrix Multiplication is equal to the Product of the Determinants ( ) BABA detdetdet ⋅= We have shown that by Invertible Elementary operations a Matrix A can be transformed to a Diagonal Matrix D. Each operation is to add to a given row one other row multiplied by a scalar (rj+α ri → rj ). According to Property (6) the value of the Determinant is unchanged by those operations. ( ) AAEDAED detdetdet ==⇒= Therefore by doing the same Elementary Operations on (A B) Matrix we have: ( ) ( )( ) ( ) BABDBDBAEBA ADDAE detdetdetdetdetdetdet detdet ⋅=⋅=== == ( ) BDBD detdetdet ⋅= q.e.d. Diagonalization of A [ ]n nnnnknn nk nk ccc r r r aaaa aaaa aaaa A       21 2 1 21 222221 111211 =             =             =
  • 98. 98 SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof mxmnxn mxmmxn nxmnxn BA BC A detdet 0 det ⋅=      14 Block Matrices Determinants q.e.d. ( ) ( )                     =              =      −− mmxn nxmn mxmmxn nxmnxn mxmmxn nxmnxn mmxn nxmn mxmmxn nxmnxn mxmmxn nxmnxn ICB I B I I A ICB I B A BC A 11 0 0 0 0 00 0 00 ( ) ( )                     =      − mmxn nxmn mxmmxn nxmnxn mxmmxn nxmnxn mxmmxn nxmnxn ICB I B I I A BC A 1 11 0 det 0 0 det 0 0 det 0 det ( ) ( ) ( ) A I A I A Laplace mxnm mnxnxn Laplace mxmmxn nxmnxn det 0 0 det1 0 0 det 11 1 ==        ⋅=      −− −  ( ) ( ) ( ) mxm LaplaceLaplace mxmnmx xmnn Laplace mxmmxn nxmnxn B B I B I det1 0 0 det1 0 0 det 1 11 ⋅==        ⋅=      − −−  ( ) 1 0 det 1 Triangular Matrix mmxn nxmn ICB I =        − mxmnxn mxmmxn nxmnxn BA BC A detdet 0 det =     
  • 99. 99 SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof [ ] [ ]    −⋅ −⋅ =      −− −− existsBifCBDAB existsAifDACBA BC DA mxnmxmnxmnxnmxm nxmnxnmxnmxmnxn mxmmxn nxmnxn 11 11 detdet detdet det 15 Block Matrices Determinants q.e.d.                −               −       =      − − − − − − existsBif ICB CBDA B DI existsAif DCAB DAI IC A BC DA m n n m mxmmxn nxmnxn 1 1 1 1 1 1 0 0 0 0 ( ) ( ) ( )    − − =                −               −       =      −− −− − − − − − − existsBifCBDAB existsAifDCABA existsBif ICB CBDA B DI existsAif DCAB DAI IC A BC DA m n n m mxmmxn nxmnxn 11 1112 1 1 1 1 1 1 detdet detdet 0 det 0 det 0 det 0 det det
  • 100. 100 SOLO Matrices Determinant of a Square Matrix – det A or |A| Proof ( ) ( )nxmmxnmmxnnxmn ABIBAI +=+ detdet 16 Block Matrices Determinants q.e.d. ( ) ( ) ( ) ( ) ( ) ( )nxmmxnmxmmxnnxmnxn nxmnxnmxnmxmmxnmxmnxmnxn mxmmxn nxmnxn ABIBAI AIBIBIAI IB AI +=+= +=+=      − −− detdet detdetdet 1 13 1 13 Sylvester's Determinant Theorem James Joseph Sylvester (1814 – 1987)
  • 101. 101 SOLO Matrices Determinant of a Square Matrix – det A or |A| 17 Cauchy - Binet Formula Jacques Philippe Marie Binet (1786 –1856) Augustin-Louis Cauchy (1789 –1857) Let A be an m×n matrix and B an n×m matrix (m≤n). Write [n] for the set { 1, ..., n }, and for the set of m-combinations of [n] (i.e., subsets of size m; there are of them). For , write A[m],S for the m×m matrix whose columns are the columns of A at indices from S, and BS,[m] for the m×m matrix whose rows are the rows of B at indices from S. The Cauchy–Binet formula then states ( ) [ ]( ) [ ]( ) [ ] ∑      ∈ = m nS mSSm BAAB ,, detdetdet                         = nmnn m m mnmm n n nxmmxn bbb bbb bbb aaa aaa aaa BA         21 22221 11211 21 22221 11211 If m=n, and we recover ( ) BAAB detdetdet =1=      n n
  • 102. 102 It was Cauchy in 1812 who used 'determinant' in its modern sense. Cauchy's work is the most complete of the early works on determinants. He reproved the earlier results and gave new results of his own on minors and adjoints. In the 1812 paper the multiplication theorem for determinants is proved for the first time although, at the same meeting of the Institut de France, Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by Cauchy. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86
  • 103. 103 SOLO Matrices Determinant of a Square Matrix – det A or |A| Example       − − = 210 121 A           − − = 03 11 12 B 4 03 12 20 11 03 11 21 12 11 12 10 21 det 323311 −= −− + − − − + − − − = −−  BA 4 17 13 det 17 13 03 11 12 210 121 −=      − −       − − =           − −       − − =BA Using Cauchy - Binet Formula we obtain: By multiplying the matrices A and B and computing det (AB), we obtain: 3 2 23 2 3 ,3,2 = ⋅ =      == nm 17 Cauchy - Binet Formula
  • 104. 104 SOLO Matrices Determinant of a Square Matrix – det A or |A| ( ) ( ) 11 detdet −− = AA18 Proof q.e.d. nIAA =−1 use ( ) ( ) ( ) ( ) ( ) AAAAAAIn det/1detdetdetdetdet1 11 11 1 1 =⇒⋅=== −−− 19 ( ) AAT detdet = Proof ( ) ( ) UDLUDLAUDLA detdetdetdetdet 11 ⋅⋅==⇒= L and U are Triangular Matrices with 1 on the Main Diagonal, and D is diagonal.  nndddDUDLAUDLA  2211 11 detdetdetdetdet ==⋅⋅=⇒= ( ) AdddLDUALDUA nn TTTTTTTT detdetdetdetdet 2211 11 11 ==⋅⋅=⇒=  q.e.d.
  • 105. 105 SOLO Matrices Determinant of the Vandermonde Matrix ( ) ( )∏≤<≤ −−− −=                 = nji ij n n nn n n xx xxx xxx xxx xxxVnxn 1 11 2 1 1 22 2 2 1 21 221 111 det,,,det       Vandermonde Matrix is a nxn Matrix that has in its j row the entries x1 j-1 x2 j-1 … xn j-1 Determinant of a Square Matrix – det A or |A| 20 Proof: Using elementary operation, let multiply the (j-1) row by –x1 and add to row j, starting with j=n, then (n-1) until j=1                 = −−− →+−→+−→+−→+−→+−→+− −−−−−−−− 11 2 1 1 22 2 2 1 21 111 11112122111111212211 n n nn n n rrrxrrrxrrrxrrrxrrrxrrrx xxx xxx xxx EEEVEEE nnnnnnnxnnnnnnn       jj x E jjj rrxr ↑↑−                     − =→− − 1 1000 010 0010 0001 1 11       ←j-1 ←j ( ) ( )1 1 1 11 det,det 22 − − − −=      = nn nn nn xx xx xxV xWe have:
  • 106. 106 SOLO Matrices Determinant of the Vandermonde Matrix Vandermonde Matrix is a nxn Matrix that has in its j row the entries x1 j-1 x2 j-1 … xn j-1 Determinant of a Square Matrix – det A or |A| Proof (continue – 1): Using fact (13) that determinant of a product of Matrices is the product of their determinants                 −− −− −− =                 −−−−−−− →+−→+−→+− −−−− 2 1 12 21 1 2 1 2 21 2 2 112 11 2 1 1 22 2 2 1 21 0 0 0 111111 1111212211 n n n n nn nn n n n nn n n rrrxrrrxrrrx xxxxxx xxxxxx xxxx xxx xxx xxx EEE nnnnnn            ( ) ( ) ( )               −− −− −− =                 −− −− −− =                 −−−− −−−−−−− →+−→+−→+− −−−− 2 1 12 21 1 2 1 2 21 2 2 112 2 1 12 21 1 2 1 2 21 2 2 112 11 2 1 1 22 2 2 1 21 111 det 0 0 0 111 det 111 detdetdetdet 1111212211 n n n n nn nn n n n n n nn nn n n n nn n n rrrxrrrxrrrx xxxxxx xxxxxx xxxx xxxxxx xxxxxx xxxx xxx xxx xxx EEE nnnnnn                       
  • 107. 107 SOLO Matrices Determinant of the Vandermonde Matrix Vandermonde Matrix is a nxn Matrix that has in its j row the entries x1 j-1 x2 j-1 … xn j-1 Determinant of a Square Matrix – det A or |A| Proof (continue – 2): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nnxnn n n n n n n n n n nn n n n nn n n nnxn xxVxxxx xx xx xxxx xxxxxx xxxxxx xxxx xxx xxx xxx xxxV ,,det 11 detdet 111 det,,,det 211112 22 2 2 112 4 1 2 12 2 2 1122 112 11 2 1 1 22 2 2 1 21 21                 −− −−−− −−− −−=               −−=               −− −− −− =                 = We obtained a recursive relation between the nxn Vandermonde Matrix V (x1, x2, … , xn) and the (n-1)x(n-1) Matrix V (x2, … ,xn), and by continuing the procedure, and because det V2x2 (xn-1,xn)=(xn-xn-1), we obtain ( ) ( )∏≤<≤ −−− −=                 = nji ij n n nn n n xx xxx xxx xxx xxxVnxn 1 11 2 1 1 22 2 2 1 21 221 111 det,,,det       q.e.d. Use Property (4) that if the elements of a row/column of the Matrix A have a common factor λ than the Determinant of A is equal to the product of λ and the Determinant of the Matrix obtained by dividing the previous row/column by λ.
  • 108. 108 SOLO Matrices Eigenvalues and Eigenvectors of Square Matrices Anxn The relation represents a Linear Transformation of the vector to . 11 nxnxnnx xAy = 1nxx 1nxy For the Square Matrix Anxn a nonzero Vector is an Eigenvector if there is a Scalar λ (called the Eigenvalue) such that: 1nxv 11 nxnxnxn vvA λ= To find the Eigenvalues and Eigenvectors we see that ( ) 01 =− nxnnxn vIA λ This equation has a solution iff the Matrix (Anxn-λ In) is singular or01 ≠nxv ( ) 0det =− nnxn IA λ This equation may be used to find the Eigenvalues λ.
  • 109. 109 SOLO Matrices Eigenvalues and Eigenvectors of Square Matrices Anxn ( ) ( ) 01det 1 1 ' 21 22221 11211 =+++−=             − − − − n nnn RuleLeibniz nnnn n n cc aaa aaa aaa      λλ λ λ λ The equation that may be used to find the Eigenvalues λ can be written as: The polynomial: ( ) ( ) ( ) ( ) ( )nn nn ccp λλλλλλλλλ −−−=+++= −  21 1 1: is called the Characteristic Polynomial of the Square Matrix Anxn, and it has degree n and therefore n Eigenvalues λ1, λ2,…, λn. However the Characteristic Equations need not have distinct solutions, there may be less than n distinct eigenvalues. If the matrix has real entries, the coefficients of the characteristic polynomial are all real. However, the roots are not necessarily real; they may include complex numbers with a non-zero imaginary component. However, there is at least one complex number λ solving the characteristic equation, even if the entries of the matrix A are complex numbers to begin with. (This existence of such a solution is known as the Fundamental Theorem of Algebra.) For a complex eigenvalue, the corresponding eigenvectors also have complex components. By Abel’s Theorem (1824) there are no algebraic formulae for the roots of a general polynomial with n > 4, therefore we need an iterative algorithm to find the roots.
  • 110. 110 SOLO Matrices Eigenvalues and Eigenvectors of Square Matrices Anxn Theorem: The n Eigenvectors of a Square Matrix Anxn that has distinct Eigenvalues are Linearly Independent. Proof: Let assume that we have k (2 ≤ k ≤ n) Linearly Dependent Eigenvvectors. Then there exist k nonzero constants αi (i=1,…,k) such that: ivvv ikkii ∀≠=++++ 0011 αααα  kivvvA iiiinxn ,,1,0 =≠= λwhere: we have: ( ) ( ) ( ) 10 0 111 11111 ≠≠−=−=− =−=− iifvvvAvIA vvAvIA iiiinxninnxn nxnnnxn λλλλ λλ ( ) ( ) ( ) ( ) ( ) 0112122111 =−++−++−=++++− kkkiiikkiinnxn vvvvvvIA λλαλλαλλααααλ  In the same way multiplying the result by (Anxn – λ2 In) we obtain: ( ) ( ) ( )[ ] ( ) ( ) ( )( ) 012213233121222 =−−++−−=−++−− kkkkkkknnxn vvvvIA λλλλαλλλλαλλαλλαλ  Continuing the procedure until, at the end, we multiply by (Anxn – λ(k-1) In) to obtain: ( )  00 0 0 1 1 =⇒=       − ≠ ≠ − = ∏ kk k i ikk v αλλα    This contradicts the assumption that αk ≠ 0 therefore the k Eigenvectors are Linearly Independent.
  • 111. 111 SOLO Matrices Eigenvalues and Eigenvectors of Square Matrices Anxn Theorem: If the n Eigenvectors of a Square Matrix Anxn corresponding to the n Eigenvalues (not necessary distinct) are Linear Independent than we can write Proof:             =Λ=− n PAP λ λ λ     00 00 00 2 1 1 Using the n Eigenvectors of a Square Matrix Anxn we can write [ ] [ ] [ ]n n nn P n vvvvvvvvvA         21 2 1 221121 00 00 00             == λ λ λ λλλ or PPA Λ= Λ=− PAP 1 q.e.d. we say that the Square Matrix Anxn is Diagonalizable. Since the n Eigenvectors of Anxn are Linear Independent P is nonsingular and we have nvvv ,,, 21  Two Square Matrices A and B that are related by A=S-1 B S are called Similar Matrices Return to Matrix Decomposition

Notes de l'éditeur

  1. http://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality
  2. http://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality http://mathworld.wolfram.com/SchwarzsInequality.html
  3. http://en.wikipedia.org/wiki/Matrix_multiplication
  4. http://en.wikipedia.org/wiki/Matrix_multiplication
  5. http://en.wikipedia.org/wiki/Matrix_multiplication
  6. http://en.wikipedia.org/wiki/Kronecker_product http://en.wikipedia.org/wiki/Leopold_Kronecker
  7. http://en.wikipedia.org/wiki/Householder_transformation http://www-history.mcs.st-andrews.ac.uk/Biographies/Householder.html G. Strang, “Linear Algebra and its Applications”, Academic Press, 2nd Ed., 198
  8. http://en.wikipedia.org/wiki/Matrix_inversion#Methods_of_matrix_inversion
  9. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86
  10. http://en.wikipedia.org/wiki/Matrix_inversion#Methods_of_matrix_inversion
  11. http://en.wikipedia.org/wiki/Matrix_inversion#Methods_of_matrix_inversion
  12. http://en.wikipedia.org/wiki/Matrix_inversion#Methods_of_matrix_inversion
  13. http://en.wikipedia.org/wiki/Matrix_inversion#Methods_of_matrix_inversion
  14. http://en.wikipedia.org/wiki/Matrix_inversion#Methods_of_matrix_inversion
  15. http://en.wikipedia.org/wiki/Matrix_inversion#Methods_of_matrix_inversion
  16. http://en.wikipedia.org/wiki/Cramer’s_rule
  17. http://en.wikipedia.org/wiki/Cramer%27s_rule
  18. http://en.wikipedia.org/wiki/Cramer%27s_rule
  19. http://en.wikipedia.org/wiki/cauchy_binet
  20. http://en.wikipedia.org/wiki/cauchy_binet
  21. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  22. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  23. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  24. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  25. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  26. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  27. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  28. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  29. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  30. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  31. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  32. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  33. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  34. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  35. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  36. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  37. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  38. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  39. http://en.wikipedia.org/wiki/Geometric_multiplicity#Algebraic_and_geometric_multiplicities
  40. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html#86
  41. http://en.wikipedia.org/wiki/Matrix_inversion#Methods_of_matrix_inversion
  42. http://en.wikipedia.org/wiki/Matrix_decomposition
  43. http://en.wikipedia.org/wiki/Matrix_decomposition
  44. http://en.wikipedia.org/wiki/Matrix_decomposition
  45. http://en.wikipedia.org/wiki/Matrix_decomposition
  46. http://en.wikipedia.org/wiki/Householder_transformation http://www-history.mcs.st-andrews.ac.uk/Biographies/Householder.html G. Strang, “Linear Algebra and its Applications”, Academic Press, 2nd Ed., 198
  47. http://en.wikipedia.org/wiki/Householder_transformation http://www-history.mcs.st-andrews.ac.uk/Biographies/Householder.html G. Strang, “Linear Algebra and its Applications”, Academic Press, 2nd Ed., 198
  48. http://en.wikipedia.org/wiki/Householder_transformation http://www-history.mcs.st-andrews.ac.uk/Biographies/Householder.html G. Strang, “Linear Algebra and its Applications”, Academic Press, 2nd Ed., 198
  49. http://en.wikipedia.org/wiki/Householder_transformation http://www-history.mcs.st-andrews.ac.uk/Biographies/Householder.html G. Strang, “Linear Algebra and its Applications”, Academic Press, 2nd Ed., 198
  50. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  51. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  52. http://www.maths.lancs.ac.uk/~gilbert/m306c/node22.html
  53. http://www.maths.lancs.ac.uk/~gilbert/m306c/node22.html
  54. http://www.maths.lancs.ac.uk/~gilbert/m306c/node22.html
  55. http://www.maths.lancs.ac.uk/~gilbert/m306c/node22.html
  56. http://www.maths.lancs.ac.uk/~gilbert/m306c/node22.html
  57. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  58. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  59. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  60. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  61. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  62. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967, pp.94-95
  63. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  64. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  65. Richard Turner, “Inverse of the Vandermonde Matrix with Applications”, NASA TN D-3547, August 1966
  66. Richard Turner, “Inverse of the Vandermonde Matrix with Applications”, NASA TN D-3547, August 1966
  67. K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967
  68. http://en.wikipedia.org/wiki/Matrix_decomposition
  69. http://en.wikipedia.org/wiki/Matrix_decomposition
  70. http://en.wikipedia.org/wiki/Matrix_decomposition