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Lecture 2 Matrices

Lecture 2 Matrices. Lat Time - Course Overview - Introduction to Systems of Linear Equations - Matrices, Gaussian Elimination and Gauss-Jordan Elimination Reading Assignment : Chapter 1 of Text Homework #1_Chap1 Assigned (Due 10/05). Elementary Linear Algebra

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Lecture 2 Matrices

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  1. Lecture 2Matrices Lat Time - Course Overview - Introduction to Systems of Linear Equations - Matrices, Gaussian Elimination and Gauss-Jordan Elimination Reading Assignment: Chapter 1 of Text Homework #1_Chap1 Assigned (Due 10/05) Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE翁慶昌-NTUEE SCC_09_2007

  2. Lecture 2: Matrices Today • Gaussian Elimination and Gauss-Jordan Elimination • Operation with Matirces • Properties of Matrix Operations Reading Assignment: Secs 2.1-2.3 of Textbook Homework #1_Chap1 Assigned (Due 10/05) Next Time • Properties of Matrix Operations • Elementary Matrices • Applications: Economics and Mgmt. and Engrg. • Determinant of a Matrix Reading Assignment: Secs 2.4, 2.5&3.1 of Textbook Homework #1 Due and #2 Assigned

  3. Lecture 2: Matrices Today • Operation with Matirces • Properties of Matrix Operations • Inverse of a Matrix

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

  5. j-th column vector • i-throw vector row matrix column matrix • Square matrix:m = n

  6. Diagonal matrix: • Trace:

  7. Ex:

  8. Equal matrix: • Ex 1: (Equal matrix)

  9. Matrix addition: • Ex 2: (Matrix addition)

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

  11. Sol: (a) (b) (c)

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

  13. Ex 4: (Find AB) Sol:

  14. = = = x A b • Matrix form of a system of linear equations:

  15. Partitioned matrices: submatrix

  16. = = = • a linear combination of the column vectors of matrix A: linear combination of column vectors of A

  17. 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

  18. 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:

  19. 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

  20. Notes: • 0m×n: the additive identity for the set of all m×n matrices • –A: the additive inverse of A • Properties of zero matrices:

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

  22. Transpose of a matrix:

  23. Sol: (a) (b) (c) • Ex: (Find the transpose of the following matrix) (a) (b) (c)

  24. Properties of transposes:

  25. Ex: is symmetric, find a, b, c? Sol: Q: Will A stay symmetric by a row exchange? • Symmetric matrix: A square matrix A is symmetric if A = AT • Skew-symmetric matrix: A square matrix A is skew-symmetric if AT = –A

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

  27. 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)

  28. Note: • Ex 4: Sow that AB and BA are not equal for the matrices. and Sol:

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

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

  31. 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

  32. 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

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

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

  35. Thus

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

  37. Sol: • Ex 3: (Find the inverse of the following matrix)

  38. Check: So the matrix A is invertible, and its inverse is

  39. Power of a square matrix:

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

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

  42. 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.

  43. Pf: ( A is nonsingular) • 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 (Left cancellation property) This solution is unique.

  44. Note:

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

  46. 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n matrix is called an elementary matrix if it can be obtained • from the identity matrixI by a single elementary operation. • Note: • Only do a single elementary row operation.

  47. Ex 1: (Elementary matrices and nonelementary matrices)

  48. 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.

  49. 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:

  50. row-echelon form

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