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Chapter 2 Matrices

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.

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Chapter 2 Matrices

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  1. Chapter 2Matrices 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 投影片設計編製者 R. Larsen et al. (6 Edition) 淡江大學 電機系 翁慶昌 教授

  2. (i, j)-th entry: 2.1 Operations with Matrices Matrix: row: m column: n size: m×n Elementary Linear Algebra: Section 2.1, pp.46-47

  3. j-th column vector i-throw vector row matrix column matrix Square matrix:m = n Elementary Linear Algebra: Section 2.1, p.47

  4. Diagonal matrix: Trace: Elementary Linear Algebra: Section 2.1, Addition

  5. Ex: Elementary Linear Algebra: Section 2.1, Addition

  6. Equal matrix: Ex 1: (Equal matrix) Elementary Linear Algebra: Section 2.1, p.47

  7. Matrix addition: Ex 2: (Matrix addition) Elementary Linear Algebra: Section 2.1, p.47

  8. Matrix subtraction: Ex 3: (Scalar multiplication and matrix subtraction) Find (a) 3A, (b) –B, (c) 3A–B Scalar multiplication: Elementary Linear Algebra: Section 2.1, pp.48-49

  9. Sol: (a) (b) (c) Elementary Linear Algebra: Section 2.1, p.49

  10. Size of AB Notes: (1) A+B = B+A, (2) Matrix multiplication: where Elementary Linear Algebra: Section 2.1, p.50

  11. Ex 4: (Find AB) Sol: Elementary Linear Algebra: Section 2.1, p.50

  12. = = = x A b Matrix form of a system of linear equations: Elementary Linear Algebra: Section 2.1, p.53

  13. Partitioned matrices: submatrix Elementary Linear Algebra: Section 2.1, Addition

  14. 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 • partitionedmatrix: 分割矩陣

  15. 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: Elementary Linear Algebra: Section 2.2, pp.61-62

  16. Properties of matrix addition and scalar multiplication: 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 Elementary Linear Algebra: Section 2.2, p.61

  17. Notes: • 0m×n: the additive identity for the set of all m×n matrices • –A: the additive inverse of A Properties of zero matrices: Elementary Linear Algebra: Section 2.2, p.62

  18. Transpose of a matrix: Elementary Linear Algebra: Section 2.2, p.67

  19. Sol: (a) (b) (c) Ex 8: (Find the transpose of the following matrix) (a) (b) (c) Elementary Linear Algebra: Section 2.2, p.68

  20. Properties of transposes: Elementary Linear Algebra: Section 2.2, p.68

  21. Ex: is symmetric, finda, b, c? Sol: Symmetric matrix: A square matrix A is symmetric if A = AT Skew-symmetric matrix: A square matrix A is skew-symmetric if AT = –A Elementary Linear Algebra: Section 2.2, p.68 & p.72

  22. Note: is symmetric Pf: Ex: is a skew-symmetric, finda, b, c? Sol: Elementary Linear Algebra: Section 2.2, p.72

  23. Matrix: Three situations: Real number: ab = ba (Commutative law for multiplication) (Sizes are not the same) (Sizes are the same, but matrices are not equal) Elementary Linear Algebra: Section 2.2, Addition

  24. Note: Ex 4: Sow thatAB andBA are not equal for the matrices. and Sol: Elementary Linear Algebra: Section 2.2, p.64

  25. Real number: (Cancellation law) Matrix: (1) If C is invertible, then A = B (Cancellation is not valid) Elementary Linear Algebra: Section 2.2, p.65

  26. Sol: So But Ex 5:(An example in which cancellation is not valid) Show that AC=BC Elementary Linear Algebra: Section 2.2, p.65

  27. 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

  28. 2.3 The Inverse of a Matrix Note: A matrix that does not have an inverse is called noninvertible (or singular). Inverse matrix: Consider Then (1) A is invertible (or nonsingular) (2) B is the inverse of A Elementary Linear Algebra: Section 2.3, p.73

  29. Pf: Notes: (1) The inverse of A is denoted by Thm 2.7: (The inverse of a matrix is unique) If B andC are both inverses of the matrix A, thenB = C. Consequently, the inverse of a matrix is unique. Elementary Linear Algebra: Section 2.3, pp.73-74

  30. Sol: Find the inverse of a matrix by Gauss-Jordan Elimination: Ex 2: (Find the inverse of the matrix) Elementary Linear Algebra: Section 2.3, pp.74-75

  31. Thus Elementary Linear Algebra: Section 2.3, p.75

  32. Note: If A can’t be row reduced to I, then A is singular. Elementary Linear Algebra: Section 2.3, p.76

  33. Sol: Ex 3: (Find the inverse of the following matrix) Elementary Linear Algebra: Section 2.3, p.76

  34. Check: So the matrix A is invertible, and its inverse is Elementary Linear Algebra: Section 2.3, p.77

  35. Power of a square matrix: Elementary Linear Algebra: Section 2.3, Addition

  36. Thm 2.8：(Properties of inverse matrices) If A is an invertible matrix, k is a positive integer, and c is a scalar not equal to zero, then Elementary Linear Algebra: Section 2.3, p.79

  37. Pf: Note: • Thm 2.9: (The inverse of a product) If A andB are invertible matrices of size n, thenAB is invertible and Elementary Linear Algebra: Section 2.3, p.81

  38. 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: IfC is not invertible, then cancellation is not valid. Elementary Linear Algebra: Section 2.3, p.82

  39. Pf: ( A is nonsingular) Thm 2.11: (Systems of equations with unique solutions) IfA is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by (Left cancellation property) This solution is unique. Elementary Linear Algebra: Section 2.3, p.83

  40. Note: For square systems (those having the same number of equations as variables), Theorem 2.11 can be used to determine whether the system has a unique solution. Note: (A is an invertible matrix) Elementary Linear Algebra: Section 2.3, p.83

  41. Keywords in Section 2.3: • inverse matrix: 反矩陣 • invertible: 可逆 • nonsingular: 非奇異 • singular: 奇異 • power: 冪次

  42. Three row elementary matrices: Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of a row to another row. 2.4 Elementary Matrices • Row elementary matrix: An nnmatrix is called an elementary matrix if it can be obtained from the identity matrixIn by a single elementary operation. • Note: Only do a single elementary row operation. Elementary Linear Algebra: Section 2.4, p.87

  43. Ex 1: (Elementary matrices and nonelementary matrices) Elementary Linear Algebra: Section 2.4, p.87

  44. Notes: • Thm 2.12: (Representing elementary row operations) Let E be the elementary matrix obtained by performing an elementary row operation on Im. If that same elementary row operation is performed on an mn matrixA, then the resulting matrix is given by the product EA. Elementary Linear Algebra: Section 2.4, p.89

  45. Ex 2: (Elementary matrices and elementary row operation) Elementary Linear Algebra: Section 2.4, p.88

  46. Ex 3: (Using elementary matrices) Find a sequence of elementary matrices that can be used to write the matrix A in row-echelon form. Sol: Elementary Linear Algebra: Section 2.4, pp.89-90

  47. row-echelon form Elementary Linear Algebra: Section 2.4, pp.89-90

  48. Row-equivalent: Matrix B is row-equivalent to A if there exists a finite number of elementary matrices such that Elementary Linear Algebra: Section 2.4, p.90

  49. Notes: Thm 2.13: (Elementary matrices are invertible) If E is an elementary matrix, thenexists and is an elementary matrix. Elementary Linear Algebra: Section 2.4, p.90

  50. (Elementary Matrix) (Elementary Matrix) (Elementary Matrix) • Ex: Elementary MatrixInverse Matrix Elementary Linear Algebra: Section 2.4, p.91

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