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

2. 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

3. Size of a Matrix 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 size m x n

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

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

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

7. Square Matrix a 11 a 12 a 13 … … a 1n a 21 a 22 a 23 … … a 2n … … … a ij … … a n1 a n2 a n3 … … a nn size m x n 5 21 3 40 -6 19 -8 12 50 size 3 x 3 m = n

8. Main Diagonal a 11 a 12 a 13 ……a 1n a 21 a 22 a 23 ……a 2n ………a ij …… a n1 a n2 a n3 ……a nn 5 21 3 40 -6 19 -8 12 50 5, -6, and 50 are diagonal elements i = j a11a22aij, …, …, ann

9. Symmetric Matrix a 11 a 12 a 13 ……a 1n a 21 a 22 a 23 ……a 2n ………a ij …… a n1 a n2 a n3 ……a nn a ij = a ji a12 = a21, a13 = a31, … a1n = an1

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

11. Diagonal Matrix a ij = 0, for a  j a 11 00……0 0a 22 0……0 ………a ij …… 000……0 a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

12. Diagonal Matrix 5 0 00 0 6 00 0 0 190 0 0 0 21

13. Unit or Identity Matrix 100……0 010……0 ………a ij …… 000… …1 a ij = 1, for i = j a ij = 0, for i  j a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

14. Unit or Identity Matrix 1000 0100 0010 0001 null matrix a ij =0

15. Matrix Operations

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

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

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

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

20. Multiplication of Matrices -1 52 3 6 A = B = 7 -34 –8 9 18 -43 51 C = 2 45 -69 Conformable [A] (m x n) x [B] (n x s) = [C] (m x s) A ik x B kj = C ij C ij = A i1 B 1j +a i2 B 2j + … + A in B nj Cij =  A ik B kj for k = 1 to n

21. Manual Multiplication 2 3 6 B = 4 –8 9 -1 518 -43 51 A = C = 7 -3 2 45 -69

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. a 11 a 12 a 13 x 1 P 1 a 12 a 22 a 23 x 2 = P 2 a 12 a 23 a 33 x 3 P 3 Application to Simultaneous Equations

24. 2 -5 4 x 1 44 3 1 -8 x 2 = -35 4 -7 -1 x 3 -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]

25. Inverse of a Square Matrix -2 1 A -1 = -1.50.5 Inverse of [A] = [A -1 ] [A -1 ] [A] = [I] [A] [A -1 ] = [I] 1 -2 A = 3 4

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

27. Transpose of a Matrix a ij T = a ji a 11 a 21 a 31 ……a n1 a 12 a 22 a 32 ……a n2 ………a ji …… a 1n a 2n a 3n ……a nn

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

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

30. Partitioning of Matrices A 11 ¦ A 12 A= -----¦------- A 21 ¦ A 22 B= B 11 ------ B 21

31. A 11 | A 12 A 11 B 11 +A 12 B 21 A= ---------------- AB= A 21 | A 22 A 21 B 11 +A 22 B 21 Partitioning of Matrices B 11 B =------ B 21

32. Partitioning of Matrices 19 28 A 11 B 11 = -43 34 14 -2 A 12 B 21 = 63 -9 A21B11 = [ -8 68 ] A22B21 = [ 28 -4 ]

33. 19 28 + 14 -2 -6 26 AB = -43 34 + 63 -9 = 20 25 [-8 68 ] + [28 -4] 20 64 Partitioning of Matrices A 11 B 11 +A 12 B 21 AB = A 21 B 11 +A 22 B 21

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

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

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. 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 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 Method of Successive Transformations

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

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

47. Cholesky Decomposition [A] = [L] [L]T [B] = [L] [A] = ( [L] [L] ) [A] = [B] [B] T 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 ii ij jrir 21/2

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 ij ii ir ii rj

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

51. Cholesky Decomposition l = l = l = 0 l = √2 = 1.414 l = (1-0)/1.414 = 0.707 l = (1.5-0.707 ) = 1 l = (2-0.7072)/1 = 1.5 l = (6.75–(707 +1.5 )) ½ = 2.0 12 11 21 31 22 32 33 1323 21/2 2 2

52. Cholesky Decomposition 1.414 0.0 0.0 0.707 1.0 0.0 0.707 1.5 0.0

53. Cholesky Decomposition b = b = b = 0 b = 1/1.414 = 0.707 b = -(0.707 x 0.707)/1 = -0.5 b = -(0.707x0.707+1.5(-0.5)/2 = 0.125 b =1 b = -(1.5 x 1)/2 b = 0.5 [B] 12 11 21 31 22 32 33 1323

54. Cholesky Decomposition 0.707 0.0 0.0 -0.5 1.0 0.0 0.125 -0.75 0.0 [A] = [B] [B] [A] = T

55. Cholesky Decomposition 0.77 -0.594 0.063 -0.5 1.563 -0.375 0.063 -0.375 0.25 R = K r r = K R R = {R R R R ….Rn} r = {r r r r ….rn} 1234 1234