Chapter 2A Matrices

1. Chapter 2AMatrices • 2A.1 Definition, and Operations of Matrices: 1 Sums and Scalar Products; 2 Matrix Multiplication • 2A.2 Properties of Matrix Operations; Mechanics of Matrix Multiplication • 2A.3 The Inverse of a Matrix • 2A.4 Elementary Matrices

2. 2A.1 Operations with Matrices Matrix: (i, j)-th entry: row: m column: n size: m×n

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

4. Diagonal matrix: Trace:

5. Ex:

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

7. Matrix addition: Scalar multiplication:

8. Matrix subtraction: Example: Matrix addition

9. Matrix multiplication: The i, j entry of the matrix product is the dot product of row i of the left matrix with column j of the right one.

10. Notes:(1) A+B = B+A, (2) Matrix multiplication:

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

12. Partitioned matrices: submatrix

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

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. 2A.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:

16. Properties of matrix addition and scalar multiplication: Then (1) A+B = B + A Commutative law for addition (2) A + ( B + C ) = ( A + B ) + C Associative law for addition (3) ( cd ) A = c ( dA ) Associative law for scalar multiplication Unit element for scalar multiplication (4) 1A = A (5) c( A+B ) = cA + cB Distributive law 1 for scalar multiplication (6) ( c+d ) A = cA + dA Distributive law 2 for scalar multiplication

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:

18. Properties of the identity matrix: Properties of matrix multiplication:

19. Properties of transposes: Transpose of a matrix:

20. 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

21. Note: is symmetric Pf: Ex: is a skew-symmetric, finda, b, c? Sol:

22. Matrix: Real number: ab = ba (Commutative law for multiplication) Three situations of the non-commutativity may occur : (Sizes are not the same) (Sizes are the same, but matrices are not equal)

23. Example (Case 3): Show thatAB andBA are not equal for the matrices. and Sol:

24. Real number: (Cancellation law) Matrix: (1) If C is invertible (i.e., C-1 exists), then A = B (Cancellation is not valid)

25. Sol: So But Example:An example in which cancellation is not valid Show that AC=BC C is noninvertible, (i.e., row 1 and row 2 are not independent)

26. 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

27. 2A.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

28. Pf: Notes: (1) The inverse of A is denoted by Theorem: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.

29. Sol: Find the inverse of a matrix A by Gauss-Jordan Elimination: Example: Find the inverse of the matrix

30. Thus

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

32. Sol: Example: Find the inverse of the following matrix

33. So the matrix A is invertible, and its inverse is You can check:

34. Power of a square matrix:

35. Theorem：Properties of inverse matrices If A is an invertible matrix, k is a positive integer, and c is a scalar, then

36. Pf: • Theorem:The inverse of a product If A andB are invertible matrices of size n, thenAB is invertible and Note:

37. Theorem: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.

38. Pf: ( A is nonsingular) Theorem: Systems of linear equations with unique solutions IfAis an invertible matrix, then the system of linear equations Ax = bhas a unique solution given by (Left cancellation property) This solution is unique.

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

40. 2A.4 Elementary Matrices • Row elementary matrix: An nnmatrix is called an elementary matrix if it can be obtained from the identity matrixI by a single elementary row operation. • Note: Only do a single elementary row operation.

41. Ex: (Elementary matrices and nonelementary matrices)

42. Theorem: 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. Notes:

43. Example: Using elementary matrices Find a sequence of elementary matrices that can be used to write the matrix A in row-echelon form. Sol:

44. row-echelon form

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

46. Theorem:Elementary matrices are invertible If E is an elementary matrix, thenexists and is an elementary matrix. Notes:

47. Ex: Elementary MatrixInverse Matrix

48. Theorem: 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. • 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. Pf: (2) If A is invertible,has only the trivial solution. Thus A can be written as the product of elementary matrices.

49. Ex: Find a sequence of elementary matrices whose product is Sol:

50. Note: IfA is invertible