1. = 1 -15 103103 -1 4 -2 y1y1 y2y2 y3y3 1 -15 103103103103 -1 4 -2 3 2 3 2 1 5 y 1 = 3+ 1+ 10= 14 xxx Calculate y 1 : ROW 1 Matrix-Vector multiplication

2. = 1 -15 103103 -1 4 -2 y2y2 y3y3 1 -15 103103103103 -1 4 -2 3 2 3 2 1 5 y 1 = 3+ 1+ 10= 14 x Calculate y 1 : ROW 1 14 Matrix-Vector multiplication

3. = 1 -15 103103 -1 4 -2 y2y2 y3y3 1 -15 103103103103 -1 4 -2 3 2 3 2 1 0 3 y 2 = 3+ 0+ 6= 9 xxx 14 Calculate y 2 : ROW 2 Matrix-Vector multiplication

4. = 1 -15 103103 -1 4 -2 y3y3 1 -15 103103103103 -1 4 -2 3 2 3 2 1 0 3 y 2 = 3+ 0+ 6= 9 x 14 Calculate y 2 : ROW 2 9 Matrix-Vector multiplication

5. = 1 -15 103103 -1 4 -2 y3y3 1 -15 103103103103 -1 4 -2 3 2 3 2 4 -2 y 3 = -3- 4 = -11 xxx 14 Calculate y 3 : ROW 3 9 Matrix-Vector multiplication

6. = 1 -15 103103 -1 4 -2 1 -15 103103103103 -1 4 -2 3 2 3 2 4 -2 y 3 = -3- 4 = -11 x 14 Calculate y 3 : ROW 3 9 -11 Matrix-Vector multiplication

7. = 1 -15 103103 -1 4 -2 1 -15 103103103103 -1 4 -2 3 2 14 9 -11 ANSWER: Matrix-Vector multiplication

8. 2 -30 761761 2 -2 1 3 4 = c 11 c 21 c 12 c 22 2 x 3 3 x 2 Inner sizes match so we can multiply Outer sizes are 2x2 so we get a 2x2 matrix Matrix-Matrix multiplication

9. = 2 -30 761761 c 11 2 -2 1 2 1 2 -3 0 c 11 = 4+ 6+ 0= 10 xxx Calculate c 11 : 3 4 -2 c 21 c 12 c 22 ROW 1 COLUMN 1 Matrix-Matrix multiplication

10. = 2 -30 761761 2 -2 1 2 1 2 -3 0 c 11 = 4+ 6+ 0= 10 x Calculate c 11 : 3 4 -2 c 21 c 12 c 22 ROW 1 COLUMN 1 10 Matrix-Matrix multiplication

11. = 2 -30 761761 2 -2 1 3 4 2 -3 0 c 12 = 6- 12+ 0= -6 xxx Calculate c 12 : 3 4 -2 c 21 c 12 c 22 ROW 1 COLUMN 2 10 Matrix-Matrix multiplication

12. = 2 -30 761761 2 -2 1 3 4 2 -3 0 c 12 = 6- 12+ 0= -6 x Calculate c 12 : 3 4 -2 c 21 c 22 ROW 1 COLUMN 2 10 -6 Matrix-Matrix multiplication

13. = 2 -30 761761 2 -2 1 2 1 7 6 1 c 21 = 14 - 12+ 1= 3 xxx Calculate c 21 : 3 4 -2 c 21 c 22 ROW 2 COLUMN 1 10 -6 Matrix-Matrix multiplication

14. = 2 -30 761761 2 -2 1 2 1 7 6 1 c 21 = 14 - 12+ 1= 3 x Calculate c 21 : 3 4 -2 c 22 ROW 2 COLUMN 1 10 -6 3 Matrix-Matrix multiplication

15. = 2 -30 761761 2 -2 1 3 4 7 6 1 c 22 = 21 + 24- 2= 43 xxx Calculate c 22 : 3 4 -2 c 22 ROW 2 COLUMN 2 10 -6 3 Matrix-Matrix multiplication

16. = 2 -30 761761 2 -2 1 3 4 7 6 1 c 22 = 21 + 24- 2= 43 x Calculate c 22 : 3 4 -2 ROW 2 COLUMN 2 10 -6 3 43 Matrix-Matrix multiplication

17. = 2 -30 761761 2 -2 1 3 4 10 -6 3 43 ANSWER: Matrix-Matrix multiplication

18. 2 -30 761761 2 -2 1 3 4 2 x 3 3 x 2 Inner sizes match so we can multiply Outer sizes are 3x3 so we get a 3x3 matrix = c 11 c 21 c 12 c 22 c 31 c 23 c 13 c 33 c 32 Multiply the other way:

19. 2 -30 761761 2 -2 1 3 4 = c 11 c 21 c 12 c 22 c 31 c 23 c 13 c 33 c 32 Calculate c 11 : ROW 1 2 3 COLUMN 1 2 7 x c 11 = 4 + 21 x = 25 Multiply the other way:

20. 2 -30 761761 2 -2 1 3 4 = c 21 c 12 c 22 c 31 c 23 c 13 c 33 c 32 Calculate c 11 : ROW 1 2 3 COLUMN 1 2 7 c 11 = 4 + 21 x = 25 25 Multiply the other way:

21. 2 -30 761761 2 -2 1 3 4 = 25 c 21 c 22 c 31 c 23 c 13 c 33 c 32 Calculate c 12 : ROW 1 2 3 COLUMN 2 -3 6 x c 12 = -6 + 18 x = 12 c 12 Multiply the other way:

22. 2 -2 1 3 4 = 25 c 21 12 c 22 c 31 c 23 c 13 c 33 c 32 ROW 1 2 3 -3 6 c 12 = -6 + 18 x = 12 Calculate c 12 : 2 -30 761761 COLUMN 2 Multiply the other way:

23. 2 -30 761761 2 -2 1 3 4 = 25 c 21 12 c 22 c 31 c 23 c 13 c 33 c 32 Calculate c 13 : ROW 1 2 3 COLUMN 3 0 1 x c 13 = 0 + 3 x = 3 Multiply the other way:

24. 2 -30 761761 2 -2 1 3 4 = 25 c 21 12 c 22 c 31 c 23 3 c 33 c 32 Calculate c 13 : ROW 1 2 3 COLUMN 3 0 1 c 13 = 0 + 3 x = 3 Multiply the other way:

25. 2 -30 761761 2 -2 1 3 4 = 2512 c 22 c 31 c 23 3 c 33 c 32 Calculate c 21 : ROW 2 -2 4 COLUMN 1 2 7 x c 21 = -4 + 28 x = 24 c 21 Multiply the other way:

26. 2 -30 761761 2 -2 1 3 4 = 25 24 12 c 22 c 31 c 23 3 c 33 c 32 Calculate c 21 : ROW 2 -2 4 COLUMN 1 2 7 c 21 = -4 + 28 x = 24 Multiply the other way:

27. 2 -30 761761 2 -2 1 3 4 = 25 24 12 c 22 c 31 c 23 3 c 33 c 32 Calculate c 22 : ROW 2 COLUMN 2 Multiply the other way:

28. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 c 31 c 23 3 c 33 c 32 Calculate c 22 : ROW 2 COLUMN 2 Multiply the other way:

29. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 c 31 c 23 3 c 33 c 32 Calculate c 23 : ROW 2 COLUMN 3 Multiply the other way:

30. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 c 31 4 3 c 33 c 32 Calculate c 23 : ROW 2 COLUMN 3 Multiply the other way:

31. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 c 31 4 3 c 33 c 32 Calculate c 31 : ROW 3 COLUMN 1 Multiply the other way:

32. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 -12 4 3 c 33 c 32 Calculate c 31 : ROW 3 COLUMN 1 Multiply the other way:

33. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 -12 4 3 c 33 c 32 Calculate c 32 : ROW 3 COLUMN 2 Multiply the other way:

34. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 -12 4 3 c 33 -15 Calculate c 32 : ROW 3 COLUMN 2 Multiply the other way:

35. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 -12 4 3 c 33 -15 Calculate c 33 : ROW 3 COLUMN 3 Multiply the other way:

36. 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 -12 4 3 -2-15 Calculate c 33 : ROW 3 COLUMN 3 Multiply the other way:

37. So: 2 -30 761761 2 -2 1 3 4 = 25 24 12 30 -12 4 3 -2-15 But: = 2 -30 761761 2 -2 1 3 4 10 -6 3 43

38. If we have a (n x m) matrix A then we can only multiply it by a (m x p) matrix B The result A B is a (n x p) matrix C (e.g. 2x3 multipled by 3x2 equals a 2x2 matrix) Found that A B does not equal B A Conclusion: Matrix multiplication

39. Rotation Matrices 1 0 cos(θ) sin(θ) θ cos(θ) sin(θ) = cos(θ) sin(θ) cos(θ) sin(θ) cos(θ) -sin(θ) 1 0 Notice that:

40. In fact: Given a point with co-ordinates Multiplying by the matrix: Is a rotation anticlockwise through an angle θ x y cos(θ) sin(θ) cos(θ) -sin(θ) x y cos(θ)x – sin(θ)y sin(θ)x + cos(θ)y =

41. x y

42. θ x y R = cos(θ) sin(θ) cos(θ) -sin(θ) R

43. 2θ2θ x y R = cos(θ) sin(θ) cos(θ) -sin(θ) R2R2

44. 3θ3θ x y R = cos(θ) sin(θ) cos(θ) -sin(θ) R3R3

45. 4θ4θ x y R = cos(θ) sin(θ) cos(θ) -sin(θ) R4R4

46. 5θ5θ x y R = cos(θ) sin(θ) cos(θ) -sin(θ) R5R5

47. a 11 a 22 a 33 a nn 0 0 00 0 0 0 00 0 0 0 1. Diagonal Matrix: Zeros everywhere apart from the leading diagonal For example: 01 0-2 00 0 0 00 0 1 0 0 0 0 3 3.6 Square Matrix Properties

48. a 11 a 22 a 33 a nn 2. Upper Triangular Matrix: Zeros below the leading diagonal. For example: 1 1 0 1 35 0 7 91 0 58 0 0 2 3 67 a 12 a 13 a 23 a 1n a 2n 3.6 Square Matrix Properties 0 0 0 0 0 0

49. 2. Lower Triangular Matrix: Zeros above the leading diagonal. For example: 0 1 1 1 30 3 7 90 1 58 3 5 0 0 67 3.6 Square Matrix Properties a 11 a 22 a 33 a nn 0 0 a 21 a 31 0 a 32 a n1 0 0 a n2 0

50. a 11 a 22 a 33 a nn 2. Symmetric Matrix: Same entries above the diagonal as below it. For example: 0 1 0 1 35 5 7 91 1 58 2 3 2 3 67 a 12 a 13 a 12 a 13 a 23 a 1n a 2n 3.6 Square Matrix Properties

51. 0 0 0 0 3. Anti-Symmetric Matrix: Zeros on leading diagonal, negative below as above. For example: 1 0 0 0-5 5 0 01 0 2 7 -2 -7 0 a 12 a 13 a 1n a 23 a 2n -a 12 -a 13 -a 1n -a 23 -a 2n 3.6 Square Matrix Properties

52. Given a matrix A, its transpose A T is found by replacing each column by the respective row. For example: 12 4 5 7 8 3 6 9 = 1 2 3 4 5 6 7 8 9 T 3.7 Transposing a matrix

53. Given a matrix A, its transpose A T is found by replacing each column by the respective row. For example: 12 4 5 3 6 = 1 2 3 4 5 6 T 3.7 Transposing a matrix

54. a 11 a 22 a 33 a mn a 12 a 13 a 21 a 31 a 23 a m1 a 32 a 2n a m2 a 11 a 22 a 33 a mn a 21 a 31 a 12 a 13 a 32 a 1n a 23 a m2 a 2n a m1 a 1n = T If A = (a ij ) then its transpose is A T = (a ji ) If A is mxn then A T is nxm 3.7 Transposing a matrix

55. 1 0 2 A = 2 4 3 0 10 2, A T = 23 4 0 ; B =, B T = A B = 1 0 2 2 4 3 0 =. So,(A B)T=(A B)T= 6 4 0 Also, 23 4 0 10 2 B T A T = = 4 6 0 6 4 0 = (A B) T 3.7.1 Transposing a product

56. In fact, if it is possible to multiply matrices A and B together then This worked for the last example, but to prove it: Let A= (a ij ) and B = (b ij ). Then B T A T = (A B) T 3.7.1 Transposing a product (B T A T ) ij = b ki a jk = a jk b ki =(A B) ji = (A B) ij T

57. If A is a square matrix, then A + A T is a symmetric matrix Proof: Let A= (a ij ) then A T = (a ji ). So, This is the same with i and j interchanged. Hence, A + A T is symmetric 1 0 2 A = 1 0 2 A + A T = 10 2 + 2 4 = A + A T = (a ij )+(a ji ) = (a ij +a ji ) 3.7.2 More Transpose properties

58. Adding transposes: Taking the transpose twice gets back to the original matrix: A T + B T = (A+B) T (A T ) T = A

59. 3.7.3 Scalar product as transpose a1a1 a2a2 a3a3 The scalar product of two vectors is: a = b1b1 b2b2 b3b3 b =a.b =a 1 b 1 +a 2 b 2 +a 3 b 3 Now treat vectors as matrices and look at: a T b =b1b1 b2b2 b3b3 a2a2 a3a3 a1a1 = a.b

60. To multiply a matrix by a scalar, you multiply every entry in the matrix by that scalar. For example: In general, if λ is a scalar, λA = (λa ij ): e.g. 1 1 1 2 A = 3 3 3 6 3A = 10 20 10A = a b c d λ λaλa λb λb λcλc λd λd = 3.8 Multiplying by a scalar

61. For each number n, the identity matrix I n is an nxn matrix with 1’s on the leading diagonal and 0’s everywhere else. E.g. 1 0 0 1 I 2 = 10 0 1 0 0 0 0 1 I 3 = 0 0 0 1 10 0 1 0 0 0 0 1 0 0 0 I 4 = 3.9 The identity matrix

62. If it is obvious (or we don’t want to specify) what size matrices we are considering we will just write “ I “ for the identity matrix I n. I has the following properties: 1.A I n = I n A for any nxn matrix A 2.I = I 2 = I 3 = I 4 = I 5 =... E.g. 1 0 0 1 1 0 0 1 1 0 0 1 = = =I2I2 I 3.9 The identity matrix