Matrix Arithmetic and Properties
Some Special Matrices
Nonsingular Matrices
Symmetric Matrices
Orthogonal and Orthonormal Matrix
Complex Matrices
Hermitian Matrices
1. MATRICES
Dr. Gabriel Obed Fosu
Department of Mathematics
Kwame Nkrumah University of Science and Technology
Google Scholar: https://scholar.google.com/citations?user=ZJfCMyQAAAAJ&hl=en&oi=ao
ResearchGate ID: https://www.researchgate.net/profile/Gabriel_Fosu2
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2. Lecture Outline
1 Introduction
Matrix Arithmetic and Properties
2 Some Special Matrices
Nonsingular Matrices
Symmetric Matrices
Orthogonal and Orthonormal Matrix
3 Complex Matrices
Hermitian Matrices
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3. Introduction Matrix Arithmetic and Properties
Definition
1 A matrix is a rectangular array of numbers, symbols, or anything with m rows and n columns
which is used to represent a mathematical object or a property of such an object. The symbol
Rm×n
denotes the collection of all m × n matrices whose entries are real numbers. Matrices
represent linear maps.
2 Matrices will usually be denoted by capital letters, and the notation A = [ai j ] specifies that the
matrix is composed of entries ai j located in the ith row and jth column of A. Example of 2×3
matrix
A =
·
1 2 −9
3 −3 2
¸
3 A vector is a matrix with either one row or one column.
Column vector (3×1):
x =
1
−3
7
Row vector (1×4):
y =
£
6 1 0 −12
¤
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4. Introduction Matrix Arithmetic and Properties
Square and Zero Matrices
Definition (Square matrix)
Matrices of size (n,n) are called square matrices or n-square matrices of order n. Examples of
2×2 and 3×3 matrices are respectively given as
·
1 2
3 4
¸
,
1 2 3
4 5 6
7 8 8
Definition (The zero matrix)
Each m ×n matrix, all of whose elements are zero, is called the zero matrix (of size m ×n) and is
denoted by the symbol 0.
S =
·
0 0
0 0
¸
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5. Introduction Matrix Arithmetic and Properties
Identity Matrix
Definition (The identity matrix)
The n × n matrix I = [δi j ], defined by δi j = 1 if i = j, and δi j = 0 if i ̸= j, is called the n × n identity
matrix of order n.
Example of 3 × 3 identity matrix is
I =
1 0 0
0 1 0
0 0 1
When any n ×n matrix A is multiplied by the identity matrix, either on the left or the right, the result
is A. Thus, the identity matrix acts like 1 in the real number system.
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6. Introduction Matrix Arithmetic and Properties
Definition (Equality of matrices)
Matrices A and B are said to be equal if they have the same size and their corresponding elements
are equal; i.e., A and B have dimension m × n, and A = [ai j ],B = [bi j ], with ai j = bi j for 1 ≤ i ≤
m, 1 ≤ j ≤ n.
Definition (Addition of matrices)
Let A = [ai j ] and B = [bi j ] be of the same size. Then A + B is the matrix obtained by adding
corresponding elements of A and B; that is,
A +B = [ai j ]+[bi j ] = [ai j +bi j ]
Definition (Scalar multiple of a matrix)
Let A = [ai j ] and t be a number (scalar). Then t A is the matrix obtained by multiplying all elements
of A by t; that is,
t A = t[ai j ] = [tai j ]
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7. Introduction Matrix Arithmetic and Properties
Definition (Negative of a matrix)
Let A = [ai j ]. Then −A is the matrix obtained by replacing the elements of A by their negatives.
Definition (Subtraction of matrices)
Matrix subtraction is defined for two matrices A = [ai j ] and B = [bi j ] of the same size, in the usual
way; that is,
A −B = [ai j ]−[bi j ]
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8. Introduction Matrix Arithmetic and Properties
Example
If
A =
·
1 2
3 4
¸
,B =
·
5 6
0 −1
¸
A +B =
·
6 8
3 3
¸
A −B =
·
−4 −4
3 5
¸
2B =
·
10 12
0 −2
¸
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9. Introduction Matrix Arithmetic and Properties
Properties
The matrix operations of addition, scalar multiplication, negation and subtraction satisfy the
following laws of arithmetic. Let s and t be arbitrary scalars and A,B,C be matrices of the same
size
1 (A +B)+C = A +(B +C)
2 A +B = B + A
3 0+ A = A
4 A +(−A) = 0
5 (s + t)A = sA + t A, (s − t)A = sA − t A
6 t(A +B) = t A + tB, t(A −B) = t A − tB
7 s(t A) = (st)A
8 1A = A, 0A = 0, (−1)A = −A
9 t A = 0 =⇒ t = 0 or A = 0
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10. Introduction Matrix Arithmetic and Properties
Matrix Product
Let A = [ai j ] be a matrix of size m × p and B = [bjk] be a matrix of size p ×n (i.e., the number of
columns of A equals the number of rows of B). Then the product AB is an m ×n matrix.
That is, if
A =
·
a b
c d
¸
,B =
·
e f
g h
¸
then
AB =
·
ae +bg a f +bh
ce +dg c f +dh
¸
Example
·
1 2
3 4
¸·
5 6
7 8
¸
=
·
19 22
43 50
¸
£
3 4
¤
·
1
2
¸
=
£
11
¤
Matrix multiplication is not commutative AB ̸= B A
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11. Introduction Matrix Arithmetic and Properties
Trace
Definition (Trace)
If A is an n × n matrix, the trace of A, written trace(A), is the sum of the main diagonal elements;
that is,
trace(A) = a11 + a22 +···+ ann =
n
X
i=1
aii
Example
If
A =
·
1 2
3 4
¸
,B =
·
5 6
0 −1
¸
then
trace(A) = 1+4 = 5
trace(B) = 5+(−1) = 4
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12. Introduction Matrix Arithmetic and Properties
Properties of Trace
trace(A +B) = trace(A)+ trace(B)
trace(c A) = c · trace(A), where c is a scalar.
trace(AB) = trace(B A)
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13. Introduction Matrix Arithmetic and Properties
Power of a Matrix
Definition (kth power of a matrix)
If A is an n ×n matrix, we define Ak
as follows:
A0
= I
and
Ak
= A × A × A··· A × A; A occurs k times for k ≥ 1.
Example
A4
= A × A × A × A
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14. Introduction Matrix Arithmetic and Properties
The transpose of a matrix
Definition (The transpose of a matrix)
Let A be an m ×n matrix. Then AT
, the transpose of A, is the matrix obtained by interchanging the
rows and columns of A. In other words if A = [ai j ], then (AT
)i j = aji . If
A =
·
1 2 3
0 −1 15
¸
then
AT
=
1 0
2 −1
3 15
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15. Introduction Matrix Arithmetic and Properties
Properties of Transpose
1 (AT
)T
= A
2 (A ±B)T
= AT
±BT
if A and B are m ×n
3 (sA)T
= sAT
if s is a scalar
4 (AB)T
= BT
AT
if A is m ×k and B is k ×n
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16. Some Special Matrices
Definition (Diagonal Matrix)
The aii ,1 ≤ i ≤ n, entries of a square matrix are called the diagonal elements. If the nondiagonal
elements are all zero, then the matrix is called a diagonal matrix. It is denoted by A =
diag(a11,a22,...,ann). Some examples are
·
10 0
0 −13
¸
,
14 0 0
0 −9 0
0 0 3
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17. Some Special Matrices
Definition (Bidiagonal Matrix)
A bidiagonal matrix is a matrix with nonzero entries along the main diagonal and either the diagonal
above or the diagonal below.
The matrix B1 is an upper bidiagonal matrix.
5 1 0 0
0 10 −1 0
0 0 9 2
0 0 0 −2
(1)
The matrix B2 is a lower bidiagonal matrix.
5 0 0 0
2 10 0 0
0 9 9 0
0 0 −1 −2
(2)
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18. Some Special Matrices
Definition (Tridiagonal Matrix)
A tridiagonal matrix has only nonzero entries along the main diagonal and the diagonals above and
below. T is a 5×5 tridiagonal matrix.
A =
2 1 0 0 0
3 4 5 0 0
0 −1 −3 2 0
0 0 1 2 10
0 0 0 −6 7
(3)
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19. Some Special Matrices Nonsingular Matrices
Nonsingular matrix
Definition (Nonsingular matrix)
An n ×n matrix A is called nonsingular or invertible if there exists an n ×n matrix B such that
AB = B A = I
The matrix B is the inverse of A. If A does not have an inverse, A is called singular.
Properties: Let denote the inverse of A by A−1
, then
1 (A−1
)A = I = A(A−1
)
2 (A−1
)−1
= A
3 (AB)−1
= B−1
A−1
4 if A is nonsingular, then AT
is also nonsingular and (AT
)−1
= (A−1
)T
Homogeneous system
A linear system Ax = 0 is said to be homogeneous. If A is nonsingular, then x = A−1
(0) = 0, so the
system has only 0 as its solution. It is said to have only the trivial solution.
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20. Some Special Matrices Symmetric Matrices
Symmetric and Skew Symmetric matrix
Symmetric
A matrix A is symmetric if AT
= A.
Another way of looking at this is that when the rows and columns are interchanged, the resulting
matrix is A.
Skew symmetric
A matrix A is skew symmetric if AT
= −A.
Example
The matrix A =
2 −3 4
−3 1 2
4 2 3
is symmetric and B =
0 2 1
−2 0 −3
−1 3 0
is skew symmetric.
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21. Some Special Matrices Symmetric Matrices
Symmetric Definite Matrices
Definition (Symmetric Positive Definite Matrix)
A symmetric matrix A is positive definite if for every nonzero vector x =
x1
x2
.
.
.
xn
xT
Ax > 0 (4)
The expression xT
Ax is called the quadratic form associated with A.
Note
The sum of two positive definite matrices is positive definite.
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22. Some Special Matrices Symmetric Matrices
Definition (Symmetric Positive Semidefinite Matrix)
A is symmetric positive semidefinite if for every nonzero vector x =
x1
x2
.
.
.
xn
xT
Ax ≥ 0 (5)
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23. Some Special Matrices Symmetric Matrices
Example
The symmetric matrix A =
·
1 0
0 1
¸
is positive definite because for
x =
·
x1
x2
¸
̸=
·
0
0
¸
(6)
then
xT
Ax = [x1 x2]
·
1 0
0 1
¸·
x1
x2
¸
(7)
= x2
1 + x2
2 (8)
Since x2
1 + x2
2 > 0 then A is a symmetric positive definite matrix.
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24. Some Special Matrices Symmetric Matrices
Theorem
1 If A = (ai j ) is positive definite, then aii > 0 for all i.
2 If A = (ai j ) is positive definite, then the largest element in magnitude of all matrix entries must
lie on the diagonal.
Example
The matrix A =
1 2 3
4 0 1
2 5 6
cannot be positive definite because A has a diagonal element of 0
Example
The matrix B =
1 −1 0 9
8 45 3 19
0 15 16 35
3 −55 2 22
cannot be positive definite because the largest element in
magnitude (−55) is not on the diagonal of B.
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25. Some Special Matrices Symmetric Matrices
Theorem
Suppose that a real symmetric tridiagonal matrix
A =
b1 a1
a1 b2 a2
a2
...
...
... bn−1 an−1
an−1 bn
(9)
with diagonal entries all positive is strictly diagonally dominant, that is,
bi > |ai−1|+|ai |, 1 ≤ i ≤ n
Then A is positive definite.
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26. Some Special Matrices Symmetric Matrices
Definition (Symmetric Negative Definite Matrix)
A is symmetric negative definite if for every nonzero vector x =
x1
x2
.
.
.
xn
xT
Ax ≤ 0 (10)
In this case, −A is positive definite.
Definition (Symmetric Indefinite Matrix)
A is symmetric indefinite if xT
Ax assumes both positive and negative values.
Alternatively, a matrix is symmetric indefinite if it has both positive and negative eigenvalues.
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27. Some Special Matrices Orthogonal and Orthonormal Matrix
Definition (Orthogonal Matrix)
A matrix P is orthogonal if
PT
P = I (11)
The inverse of P is its transpose.
Example
The matrix P =
·
cosθ −sinθ
sinθ cosθ
¸
is orthogonal because
PT
P =
·
cosθ sinθ
−sinθ cosθ
¸·
cosθ −sinθ
sinθ cosθ
¸
(12)
=
·
cos2
θ +sin2
θ 0
0 cos2
θ +sin2
θ
¸
(13)
=
·
1 0
0 1
¸
(14)
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28. Some Special Matrices Orthogonal and Orthonormal Matrix
Theorem
The matrix P is orthogonal if and only if the columns of P are orthogonal and have unit length.
〈vi , vj 〉 =
(
1 if i = j
0 if i ̸= j
(15)
Example
D = [e1 e2 e3] =
1 0 0
0 1 0
0 0 1
(16)
then
〈e1, e2〉 = 〈e1, e3〉 = 〈e2, e3〉 = 0 (17)
and
〈e1, e1〉 = 〈e2, e2〉 = 〈e3, e3〉 = 1 (18)
Hence D is an orthogonal matrix.
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29. Some Special Matrices Orthogonal and Orthonormal Matrix
Definition (Orthonormal)
1 A set of orthogonal vectors, each with unit length, are said to be orthonormal.
2 D = [e1 e2 e3] is an orthogonal matrix, and each orthogonal vector e1, e2 and e2 has a unit length.
3 Hence e1, e2,··· ,en are called the standard orthonormal basis.
The 3×3 identity matrix is orthogonal, each column vectors has unit lenght, thus forms a set of
orthonormal basis
|e1| = |e2| = |e3| =
q
a2
11 + a2
21 + a2
31 =
p
12 = 1 (19)
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30. Complex Matrices Hermitian Matrices
If a = 1+3i then the complex conjugate is ā = 1−3i
Definition (Complex conjugate matrix)
The complex conjugate of an n×m complex matrix A = (zi j ) is defined and denoted by Ā = (z̄i j )m×n.
Definition (Hermitian and Skew Hermitian matrix)
A complex n-square matrix A is said to be hermitian if
ĀT
= A or z̄ji = zi j (20)
and skew hermitian if
ĀT
= −A or z̄ji = −zi j (21)
Example (M is Hermitian and N is skew Hermitian)
M =
2 1−i 0
1+i −1 i
0 −i 2
N =
i 2+i 3+2i
−2+i 3i −3i
−3+2i −3i 0
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31. Complex Matrices Hermitian Matrices
Exercises
1 Let A, B, C, D be matrices defined by
A =
3 0
−1 2
1 1
,B =
1 5 2
−1 1 0
−4 1 3
,C =
−3 −1
2 1
4 3
,D =
·
4 −1
2 0
¸
Which of the following matrices are defined? Compute those matrices which are defined. A +
B, A +C, AB,B A,CD,DC,D2
,(CT
)T
2 Rotate the line y = −x +3 60°counterclockwise about the origin.
3 Let σ1 =
¡0 1
1 0
¢
, σ2 =
¡0 −i
i 0
¢
and σ1 =
¡1 0
0 −1
¢
be the three Pauli matrices. Show that xσ1 + yσ2 +2σ3
is a hermitian matrix for any two real numbers x, y ∈ R.
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