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TYPES OF SPECIAL SQUARE MATRIX
- 1. LEC N0 04 LINEAR ALGEBRAAND ANALYTICAL GEOMETRY TYPICAL SQUARE MATRIX EXAMPES
- 3. DEFINITION: 𝐴 𝐾+1 = 𝐴 𝐾 𝐼𝑆 𝐿𝐸𝐴𝑆𝑇 𝐼𝑁𝑇𝐸𝐺𝐸𝑅
- 4. : Show that the matrix 302 923 621 A is a periodic matrix with period 2. Solution: Consider 302 923 621 302 923 621 2A 9012004602 2718180461863 181860421261 344 9109 665 THEN WE SAY THAT A IS PERODIC MATRIX OF PERIOD K
- 5. EXAMPLE OF PERIODIC MATRIX OF PERIOD 2 Now consider, 302 923 621 344 9109 665 A2A3A 936240886124 2790540201818309 1854300121012185 A 302 923 621 Since A3 = A hence A is a periodic matrix with period 2.
- 6. IDEMPOTENT MATRIX A square matrix is called idempotent if it holds 𝑨 𝟐 = 𝑨 Example Show that 321 431 422 A is an Idempotent matrix. Solution: Consider 321 431 422 321 431 422 2A 984662322 12124892432 1288864424 321 431 422 = A Since A2 = A, the given matrix A is an Idempotent Matrix.
- 7. NILPOTENT MATRIX A SQUARE MATRIX IS CALLED NILPOTENTIF 𝑨 𝒏 = 𝟎 IF 𝐧 𝐈𝐒 𝐏𝐎𝐒𝐈𝐓𝐈𝐕𝐄 𝐈𝐍𝐓𝐄𝐆𝐄𝐑 𝐓𝐇𝐄𝐍 𝐀 𝐈𝐒 𝐂𝐀𝐋𝐋𝐄𝐃 𝐍𝐈𝐋𝐏𝐎𝐓𝐄𝐍𝐓 𝐎𝐅 𝐈𝐍𝐃𝐄𝐗 𝐧 IF A IS NILPOTENT MATRIX OF ANY INDEX THEN ITS DETERMINANT IS ZERO. EXAMPLE: Show that 431 431 431 is nilpotent. Solution: Let 431 431 431 A 431 431 431 431 431 431 AA2A O 000 000 000 161241293431 161241293431 161241293431 Thus given matrix is nilpotent of index 2. REMARK: You may observe that | A | = 0.
- 8. INVOLUTORY MATRIX A square matrix is said to be involutory matrix if 𝑨 𝟐=I Example : Show that 433 434 110 A is Involutory. Solution: Consider 16123129312120 16124129412120 440330340 433 434 110 433 434 110 2A I 100 010 001 Since, A2 = I, hence A is an Involutory matrix. REMARK: If A is Involutory then A2 = I A A = I. Pre-multiplying both sides by A-1 , we get: A-1 (A.A) = A-1 . I (A-1 A) . A = A-1 IA = A-1 A = A-1 . This shows that if A is an involutory matrix, it is equal to its inverse
- 9. TRANSPOSE OF A MATRIX
- 10. EXAMPLES
- 11. PROPERTIES OF TRANSPOSE MATRICES If matrices A and B are conformable for the sum A ± B and the product AB, then (a) (A ± B)T = AT ± BT (b) AA TT (c) (kA)T = k AT , k being scalar (d)(AB)T = BT .AT Corollary: (ABC)T = CT BTAT.
- 12. SYMETRIC AND SKEW-SYMMETRIC MATRICES Let Abe a square matrix 𝐀𝐭 = 𝐀 then A is called symmetric matrix and if 𝐀𝐭 = − 𝐀 𝐭𝐡𝐞𝐧 𝐀 𝐢𝐬 𝐜𝐚𝐥𝐥𝐞𝐝 𝐬𝐤𝐞𝐰 𝐬𝐲𝐦𝐦𝐞𝐭𝐫𝐢𝐜. For example, matrices 101312 1369 1294 and 052 5-07- 2-70 respectively are symmetric and skew-symmetric matrices.
- 13. THEOREM 01 If A is square matrix, prove that A + AT is symmetric and A – AT is a skew-symmetric matrix. Proof: Let C = A + AT then: C T = (A + AT )T = AT + (AT )T = AT + A = A + AT = C. Since CT = C, hence C is symmetric. But C = A + AT, hence A + AT is symmetric. Now let D = A – AT then, DT =(A – AT )T = AT - (AT )T = AT - A = -(A – AT ) = - D. Since DT= -D, hence D is skew-symmetric. But D = A – AT, hence A – AT is skew-symmetric.
- 14. THEOREM 02 If A is skew-symmetric matrix, then its diagonal elements are zero. Proof: Given that A is a skew-symmetric matrix, hence AT = - A. This implies that aij = - aij for all i, j. This must be true for the diagonal elements as well. Thus, aii = - aii or aii + aii = 0 or 2 aii = 0 aii = 0. This proves that for a skew-symmetric matrix A the diagonal elements are zero.
- 15. THEOREM 03 If A and B are two symmetric matrices then (AB)T = A B if and only if they commute. Proof: Given that A and B are symmetric matrices, that is, AT = A and BT = B. Now let us assume that A and B commute, that is; A B = B A (1) Then by definition, (A B)T = BT AT = B A [ Note: AT = A and BT = B] = A B [By equation (1)] Thus, (A B)T = A B. This proves that AB is symmetric. Now let us assume that AB is symmetric, i.e. (AB)T = A B. BT AT = A B B A = A B [Note: AT = A and BT = B] This shows that A and B commute.
- 16. EXAMPLE Let 545 938 712 TA 597 431 582 A 052 5-07- 2-70 TAAand 101312 1369 1294 TAA We see that A + A T is a symmetric matrix and A – AT is skew-symmetric matrix. We also see that all the diagonal elements of a skew-symmetric matrix are zero
- 17. EXAMPLE Write the following matrix as sum of symmetric and skew-symmetric matrices. 361 352 421 A Solution: Given 361 352 421 A 334 652 121 A T 693 9100 302 AA T (1) 035 304 540 AAand T (2) Adding (1) and (2), we get: 035 304 540 693 9100 302 A2 matrixsymmetricSkewmatrixSymmetric 02/32/5 2/302 2/520 32/92/3 2/950 2/301 A
- 18. CONJUGATE MATRICES A complex matrix obtained by replacing its complex elements by their corresponding complex conjugates is called the conjugate and is denoted by A For example, if i370i34 8i40 i54i32 A i370i34 8i40 i54i32 A is the conjugate matrix of A.
- 19. TRANJUGATE TRANSPOSE OF CONJUGATE OF A MATRIX
- 20. HERMITIAN MATRIX 𝐴 𝑡 = 𝐴 A square matrix A for which T__ A is called Hermitian matrix. For example, the matrix 0i3i2 i21i 3i21i1 A is a Hermitian matrix because, T__ A 0i3i2 i21i 3i21i1__ A A 0i3i2 i21i 3i21i1 Hence A is Hermitian matrix
- 21. SKEW –HERMITION MATRIX 𝐴 𝑡 =-A example, if , 0i2 ii3i1 2i1i A then 0i2 ii3i1 2i1i A A 0i2 ii3i1 2i1i 0i2 ii3i1 2i1i t A
- 22. ORTHOGONAL MATRIX A square matrix A is said to be orthogonal if AT A = I = AAT Show that the matrix below is orthogonal. cosθ0sinθ 010 sinθ0θcos Solution: By definition consider, θcos0θsin- 010 θsin0θcos θcos0θsin 010 θsin-0θcos TAA I 100 010 001 θ2sinθ2cos0sinθcosθsinθcosθ 010 sinθcosθsinθcosθ0θ2sinθ2cos Similarly we can show that AT A = I. Hence given matrix is orthogonal.
- 23. EXAMPLE Prove that 122 212 221 3 1 A is orthogonal. Solution: 122 212 221 3 1T A 122 212 221 3 1 A Then I 100 010 001 9 9 900 090 009 9 1 122 212 221 122 212 221 9 1 ATA Since, AT A = I, therefore matrix A is orthogonal.
- 24. THEOREM ON ORTHOGONALITY If A and B are orthogonal matrices, then AB and BA are also orthogonal matrices. Proof: Since A and B are orthogonal matrices, we have, AAT = I and BBT = I. We have to prove that AB and BA are also orthogonal. Consider, (AB)(AB)T = AB. BT AT = A (B BT )AT = A(I) AT = AAT = I AB is orthogonal. Similarly, consider (BA)(BA)T = BA. AT BT = B (AAT )BT = B(I) BT = B BT = I BA is orthogonal.
- 25. UNITARY MATRIX Square matrix A is said to be unitary if Show that 2 i1 2 i1 2 i1 2 i1 A is a unitary matrix. Solution: 2 i1 2 i1 2 i1 2 i1 TA 2 i1 2 i1 2 i1 2 i1 A Also 2 i1 2 i1 2 i1 2 i1 TA Now i1i1 i1i1 i1i1 i1i1 2 1 2 1 2 i1 2 i1 2 i1 2 i1 2 i1 2 i1 2 i1 2 i1 AtA I 10 01 10 01 4 4 40 04 4 1 2i12i12i1i1 2i12i12i12i1 4 1 Hence A is a unitary matrix.
- 26. THEOREM ON HERMITION MATRIX
- 27. The end