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Definition of a Matrix

A ij i = row j = column. Definition of a Matrix. A [ A ]. a 11 a 12 a 13 … … a 1n a 21 a 22 a 23 … … a 2n … … … a ij … … a m1 a m2 a m3 … … a mn. Definition of a Matrix. a 11 a 12 a 13 … … a 1n

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Definition of a Matrix

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  1. Aij i = row j = column Definition of a Matrix A [ A ]

  2. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … am1 am2 am3 … … amn Definition of a Matrix

  3. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … am1 am2 am3 … … amn size m x n Size of a Matrix

  4. Size of a Matrix 5 21 3 -7 40 -6 19 23 -8 12 50 22 size 3 x 4

  5. Row Matrix [ B ] m = 1 [ 50 -3 -27 35 ]

  6. {D} n = 1 -10 33 -6 15 {-10 33 -6 15} Column Matrix

  7. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … an1 an2 an3 … … ann size m x n 5 21 3 40 -6 19 -8 12 50 size 3 x 3 Square Matrix m = n

  8. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … an1 an2 an3 … … ann 5 21 3 40 -6 19 -8 12 50 5, -6, and 50 are diagonal elements Main Diagonal i = j a11 a22 aij, …, …, ann

  9. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … an1 an2 an3 … … ann Symmetric Matrix aij = aji a12 = a21, a13 = a31, … a1n = an1

  10. Symmetric Matrix 5 21 -3 21 6 19 -8 19 50 21, -3, and 19 are off-diagonal elements

  11. Diagonal Matrix aij = 0, for a  j a11 0 0 … … 0 0 a22 0 … … 0 … … … aij … … 0 0 0 … … 0 a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

  12. 0 0 0 • 0 6 0 0 • 0 0 19 0 • 0 0 0 21 Diagonal Matrix

  13. 1 0 0 … … 0 0 1 0 … … 0 … … … aij … … 0 0 0 … … 1 Unit or Identity Matrix aij = 1, for i = j aij = 0, for i  j a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

  14. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 null matrix aij =0 Unit or Identity Matrix

  15. Matrix Operations

  16. 5 21 -3 A = 21 6 19 -8 19 50 5 21 -3 B = 21 6 19 -8 19 50 Equality A = B Aij = Bij

  17. 5 2 A = 2 6 -8 1 5 21 B = 21 6 -8 19 Addition and Subtraction [A] + [B] = [C] Aij + Bij = Cij

  18. 10 23 A+B = C = 23 12 -16 20 0 -19 A-B = C = -19 0 0 -18 Addition and Subtraction

  19. 5 2 A = 2 6 -8 1 15 6 B = 6 18 -24 3 Multiplication by ScalarScalar c, x [A] c = 3 c A = B

  20. -1 5 2 3 6A = B = 7 -3 4 –8 9 18 -43 51 C = 2 45 -69 Multiplication of Matrices Conformable [A] (m x n) x [B] (n x s) = [C] (m x s) Aik x Bkj = C ij Cij = Ai1B1j +ai2B2j+ … + AinBnj Cij =  AikBkj for k = 1 to n

  21. 2 3 6 B = 4 –8 9 -1 5 18 -43 51 A = C = 7 -3 2 45 -69 Manual Multiplication

  22. Application to Simultaneous Equations a11x1 + a12x2 + a13x3 = P1 a12x2 + a22x2 + a23x3 = P2 a12x3 + a23x2 + a33x3 = P3 2x1 – 5x2 + 4x3 = 44 3x1 + 1x2 + -8x3 = -35 4x1 – 7x2 – 1x3 = 28

  23. Application to Simultaneous Equations a11 a12 a13 x1 P1 a12 a22 a23 x2 = P2 a12 a23 a33 x3 P3

  24. 2 -5 4 x1 44 3 1 -8 x2 = -35 4 -7 -1 x3 -28 [A] {x} = {P} NOTES: [A] [B]  [B] [A] A B C = (AB) C = A (BC) A (B + C) = AB + AC [A] [0] = [0], [0] [A] = [0] Application to Simultaneous Equations

  25. 1 -2 A = 3 4 -2 1 A-1 = -1.5 0.5 Inverse of a Square Matrix Inverse of [A] = [A-1] [A-1] [A] = [I][A] [A-1] = [I]

  26. Inverse of Square Matrix 1 0 A A-1 = 0 1

  27. Transpose of a Matrix aijT = aji a11 a21 a31 … … an1 a12 a22 a32 … … an2 … … … aji … … a1n a2n a3n … … ann

  28. 5 12 -3 18 21 6 19 16 -3 15 50 17 5 21 -3 12 6 15 -3 19 50 18 16 17 Transpose of a Matrix A (3 x 4) , AT(4 x 3)

  29. 3 5 -1 ¦ 2 -2 4 7 ¦ 9 6 1 3 ¦ 4 1 8 -5 2 -3 6 7 -1 Partitioning of Matrices [A] [B]

  30. Partitioning of Matrices A11 ¦ A12 A = -----¦------- A21 ¦ A22 B = B11 ------ B21

  31. Partitioning of Matrices B11 B = ------ B21 A11 | A12 A11B11+A12B21 A= ---------------- AB= A21 | A22 A21B11+A22B21

  32. 19 28 A11B11 = -43 34 14 -2 A12B21 = 63 -9 Partitioning of Matrices A21B11 = [ -8 68 ] A22B21 = [ 28 -4 ]

  33. Partitioning of Matrices A11B11+A12B21 AB = A21B11+A22B21 19 28 + 14 -2 -6 26 AB = -43 34 + 63 -9 = 20 25 [-8 68 ] + [28 -4] 20 64

  34. Solution of Simultaneous Equations by Gauss-Jordan Method 2x1 – 5x2 + 4x3 = 44 3x1 + x2 - 9x3 = -35 4x1 – 7x2 - x3 = 28 x1 – 2.5x2 + 2x3 = 22 3x1 + x2 - 8x3 = -35 4x1 - 7x2 - x3 = 28

  35. Solution of Simultaneous Equations by Gauss-Jordan Method x1 – 2.5x2 + 2x3 = 22 8.5x2 - 14x3 = -101 3x2 - 9x3 = -60 x1 – 2.5x2 + 2x3 = 22 x2 - 1.647x3 = -11.882 3x2 - 9x3 = -60

  36. Solution of Simultaneous Equations by Gauss-Jordan Method x1 – - 2.118x3 = -7.705 x2 - 1.647x3 = -11.882 - 4.059x3 = -24.354 x1 + 2.118x3 = - 7.705 x2 - 1.647x3 = -11.882 x3 = 6 x1 = 5 x2 = -2 x3 = 6

  37. Solution of Simultaneous Equations by Gauss-Jordan Method Check: 2(5) - 5(-2) + 4(6) = 44 3(5) +1(-2) - 8(6) = -35 4(5) - 7(-2) - 1(6) = 28

  38. Matrix Inversion [A] {x} = {C} [A] [A] {x} = [A]-1 {C} [A] [A] = [I] {x} = [A] {C} [A ¦ I ] { x ¦ -C }= 0 -1 -1 -1

  39. Matrix Inversion [I ¦ B ] { x ¦ -C }= 0 {x} - [B] [C] = 0 {x} = [B] [C] [B] = [A] -1

  40. Method of Successive Transformations 2 4 3 ¦ 1 0 0 1 -2 0 ¦ 0 1 0 -1 -4 5 ¦ 0 0 1 1 2 1.5 ¦ 0.5 0 0 1 -2 0 ¦ 0 1 0 -1 -4 5 ¦ 0 0 1

  41. Method of Successive Transformations 1 2 1.5 ¦ 0.5 0 0 0 -4 -1.5 ¦ -0.5 1 0 -1 -4 5 ¦ 0 0 1 1 2 1.5 ¦ 0.5 0 0 0 -4 -1.5 ¦ -0.5 1 0 0 -2 6.5 ¦ 0.5 0 1

  42. Method of Successive Transformations 1 2 1.5 ¦ 0.5 0 0 0 1 0.375 ¦ 0.125 -0.25 0 0 -2 6.5 ¦ 0.5 0 1 1 2 1.5 ¦ 0.5 0 0 0 1 0.375 ¦ 0.125 -0.25 0 0 0 7.25 ¦ 0.75 -0.5 1

  43. Method of Successive Transformations 1 2 1.5 ¦ 0.5 0 0 0 1 0.375 ¦ 0.125 -0.25 0 0 0 1 ¦ 0.1034 -0.06897 0.1379 1 2 1.5 ¦ 0.5 0 0 0 1 0 ¦ 0.0862 -0.2241 -0.0517 0 0 1 ¦ 0.1034 -0.06897 0.1379

  44. Method of Successive Transformations 1 2 0 ¦ 0.3449 0.1034 -0.2069 0 1 0 ¦ 0.0862 -0.2241 -0.0517 0 0 1 ¦ 0.1034 -0.06897 0.1379 1 0 0 ¦ 0.1725 0.5516 -0.1035 0 1 0 ¦ 0.0862 -0.2241 -0.0517 0 0 1 ¦ 0.1034 -0.06897 0.1379

  45. Method of Successive Transformations 0.1725 0.5516 - 0.1035 A-1 = 0.0862 - 0.2241 - 0.0517 0.1034 - 0.06897 0.1379

  46. Cholesky Decomposition Lower Triangular matrix [L] l11 0 0 . . . . 0 l21 l22 0 . . . . 0 l31 l32 l33 0 . . . 0 . . . . . . . . . . . . . . . . ln1 . . . . . . lnn

  47. Cholesky Decomposition [A] = [L] [L]T [B] = [L] [A] = ( [L] [L] ) [A] = [B] [B] -1 -1 T -1 -1 T

  48. Cholesky Decomposition Elements of [L]: l = 0 for i<j l = (A - ∑l ) l = (A - ∑l l )/l for i>j Summation ∑ from r=1 to j-1 ij 2 1/2 ir ij ii ij ij ir jr

  49. Cholesky Decomposition Elements of [B]: b = 0 for i<j b = 1/l b = -(∑l l )/l or i>j Summation ∑ from r=1 to i-1 ij ii ii ij rj ii ir

  50. Cholesky Decomposition Example: 2 1 1 1 1.5 2 1 2 6.75

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