1. Chapter 2 Matrices 2.1 Operations with Matrices 2.2 Properties of Matrix Operations 2.3 The Inverse of a Matrix 2.4 Elementary Matrices Elementary Linear Algebra 投影片設計製作者 R. Larsen et al. (5 Edition) 淡江大學 電機系 翁慶昌 教授

2. 2 - 2 2.1 Operations with Matrices Matrix: (i, j)-th entry: row: m column: n size: m×n

3. 2 - 3 i-th row vector j-th column vector row matrix column matrix Square matrix: m = n

4. 2 - 4 Diagonal matrix: Trace:

5. 2 - 5 Ex:

6. 2 - 6 Equal matrix: Ex 1 : (Equal matrix)

7. 2 - 7 Matrix addition: Ex 2: (Matrix addition)

8. 2 - 8 Matrix subtraction: Scalar multiplication: Ex 3: (Scalar multiplication and matrix subtraction) Find (a) 3A, (b) –B, (c) 3A – B

9. 2 - 9 (a)(a) (b) (b) (c)(c) Sol:

10. 2 - 10 Matrix multiplication: where Notes: (1) A+B = B+A, (2) Size of AB

11. 2 - 11 Ex 4: (Find AB) Sol:

12. 2 - 12 Matrix form of a system of linear equations: === A x b

13. 2 - 13 Partitioned matrices: submatrix

14. 2 - 14 a linear combination of the column vectors of matrix A: linear combination of column vectors of A ===

15. 2 - 15 Keywords in Section 2.1: row vector: 列向量 column vector: 行向量 diagonal matrix: 對角矩陣 trace: 跡數 equality of matrices: 相等矩陣 matrix addition: 矩陣相加 scalar multiplication: 純量積 matrix multiplication: 矩陣相乘 partitioned matrix: 分割矩陣

16. 2 - 16 2.2 Properties of Matrix Operations Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication Zero matrix: Identity matrix of order n:

17. 2 - 17 Then (1) A+B = B + A (2) A + ( B + C ) = ( A + B ) + C (3) ( cd ) A = c ( dA ) (4) 1A = A (5) c( A+B ) = cA + cB (6) ( c+d ) A = cA + dA Properties of matrix addition and scalar multiplication:

18. 2 - 18 Notes: (1)0 m×n : the additive identity for the set of all m×n matrices (2)–A: the additive inverse of A Properties of zero matrices:

19. 2 - 19 (1) Ａ (BC) = (AB ) C (2) Ａ (B+C) = AB + AC (3) (A+B)C = AC + BC (4) c (AB) = (cA) B = A (cB) Properties of the identity matrix: Properties of matrix multiplication:

20. 2 - 20 Transpose of a matrix:

21. 2 - 21 (b) (b)(c)(c) Sol: (a)(a) (b)(b) (c)(c) (a)(a) Ex: (Find the transpose of the following matrix)

22. 2 - 22 Properties of transposes:

23. 2 - 23 A square matrix A is symmetric if A = A T Ex: is symmetric, find a, b, c? A square matrix A is skew-symmetric if A T = –A Skew-symmetric matrix: Sol: Symmetric matrix:

24. 2 - 24 is a skew-symmetric, find a, b, c? Note: is symmetric Pf: Sol: Ex:

25. 2 - 25 ab = ba(Commutative law for multiplication) (Sizes are not the same) (Sizes are the same, but matrices are not equal) Real number: Matrix: Three situations:

26. 2 - 26 Sol: Note: Ex 4 : Sow that AB and BA are not equal for the matrices. and

27. 2 - 27 (Cancellation is not valid) (Cancellation law) Matrix: (1) If C is invertible, then A = B Real number:

28. 2 - 28 Sol: So But Ex 5: (An example in which cancellation is not valid) Show that AC=BC

29. 2 - 29 Keywords in Section 2.2: zero matrix: 零矩陣 identity matrix: 單位矩陣 transpose matrix: 轉置矩陣 symmetric matrix: 對稱矩陣 skew-symmetric matrix: 反對稱矩陣

30. 2 - 30 2.3 The Inverse of a Matrix Note: A matrix that does not have an inverse is called noninvertible (or singular). Consider Then (1) A is invertible (or nonsingular) (2) B is the inverse of A Inverse matrix:

31. 2 - 31 If B and C are both inverses of the matrix A, then B = C. Pf: Consequently, the inverse of a matrix is unique. Notes: (1) The inverse of A is denoted by Thm 2.7: (The inverse of a matrix is unique)

32. 2 - 32 Ex 2: (Find the inverse of the matrix) Sol: Find the inverse of a matrix by Gauss-Jordan Elimination:

33. 2 - 33 Thus

34. 2 - 34 If A can’t be row reduced to I, then A is singular. Note:

35. 2 - 35 Sol: Ex 3: (Find the inverse of the following matrix)

36. 2 - 36 So the matrix A is invertible, and its inverse is Check:

37. 2 - 37 Power of a square matrix:

38. 2 - 38 If A is an invertible matrix, k is a positive integer, and c is a scalar, then Thm 2.8 ： (Properties of inverse matrices)

39. 2 - 39 Thm 2.9: (The inverse of a product) If A and B are invertible matrices of size n, then AB is invertible and Pf: Note:

40. 2 - 40 Thm 2.10 (Cancellation properties) If C is an invertible matrix, then the following properties hold: (1) If AC=BC, then A=B (Right cancellation property) (2) If CA=CB, then A=B (Left cancellation property) Pf: Note: If C is not invertible, then cancellation is not valid.

41. 2 - 41 Thm 2.11: (Systems of equations with unique solutions) If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by Pf: ( A is nonsingular) This solution is unique. (Left cancellation property)

42. 2 - 42 Note:

43. 2 - 43 Keywords in Section 2.3: inverse matrix: 反矩陣 invertible: 可逆 nonsingular: 非奇異 singular: 奇異 power: 冪次

44. 2 - 44 2.4 Elementary Matrices Row elementary matrix: An n  n matrix is called an elementary matrix if it can be obtained from the identity matrix I by a single elementary operation. Three row elementary matrices: Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of a row to another row. Note: Only do a single elementary row operation.

45. 2 - 45 Ex 1: (Elementary matrices and nonelementary matrices)

46. 2 - 46 Notes: Thm 2.12: (Representing elementary row operations) Let E be the elementary matrix obtained by performing an elementary row operation on I m. If that same elementary row operation is performed on an m  n matrix A, then the resulting matrix is given by the product EA.

47. 2 - 47 Sol: Ex 3: (Using elementary matrices) Find a sequence of elementary matrices that can be used to write the matrix A in row-echelon form.

48. 2 - 48 row-echelon form

49. 2 - 49 Matrix B is row-equivalent to A if there exists a finite number of elementary matrices such that Row-equivalent:

50. 2 - 50 Thm 2.13 : (Elementary matrices are invertible) If E is an elementary matrix, then exists and is an elementary matrix. Notes:

51. 2 - 51 Ex: Elementary Matrix Inverse Matrix

52. 2 - 52 Pf: (1)Assume that A is the product of elementary matrices. (a) Every elementary matrix is invertible. (b) The product of invertible matrices is invertible. Thus A is invertible. (2) If A is invertible, has only the trivial solution. (Thm. 2.11) Thus A can be written as the product of elementary matrices. Thm 2.14: (A property of invertible matrices) A square matrix A is invertible if and only if it can be written as the product of elementary matrices.

53. 2 - 53 Sol: Ex 4: Find a sequence of elementary matrices whose product is

54. 2 - 54 Note: If A is invertible

55. 2 - 55 If A is an n  n matrix, then the following statements are equivalent. (1) A is invertible. (2) Ax = b has a unique solution for every n  1 column matrix b. (3) Ax = 0 has only the trivial solution. (4) A is row-equivalent to I n. (5) A can be written as the product of elementary matrices. Thm 2.15: (Equivalent conditions)

56. 2 - 56 L is a lower triangular matrix U is an upper triangular matrix If the n  n matrix A can be written as the product of a lower triangular matrix L and an upper triangular matrix U, then Note: If a square matrix A can be row reduced to an upper triangular matrix U using only the row operation of adding a multiple of one row to another, then it is easy to find an LU-factorization of A. LU-factorization:

57. 2 - 57 Sol: (a) Ex 5: (LU-factorization)

58. 2 - 58 (b)

59. 2 - 59 Two steps: (1) Write y = Ux and solve Ly = b for y (2) Solve Ux = y for x Solving Ax=b with an LU-factorization of A

60. 2 - 60 Sol: Ex 7: (Solving a linear system using LU-factorization)

61. 2 - 61 Thus, the solution is So

62. 2 - 62 Keywords in Section 2.4: row elementary matrix: 列基本矩陣 row equivalent: 列等價 lower triangular matrix: 下三角矩陣 upper triangular matrix: 上三角矩陣 LU-factorization: LU 分解