1. Lecture 3Matrices Lat Time Matrices, Gaussian Elimination and Gauss-Jordan Elimination Operation with Matirces Properties of Matrix Operations Reading Assignment: Sec. 1.1 and 2.1of Text Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE翁慶昌-NTUEE SCC_09_2007

2. Application: Model/Curve Fitting • Problem Description • A baseball analyst says that “Wang’s strike out rate (SO) is parabolic function of his average pitch speed (v)” • You are suspicious of the analyst’s theory. So you collect some data about Wang’s pitching performance as follows: • Now you want to find out if a parabolic function is a good model of SO = f(v)

3. Ideas • I1: Mathematical formulation of the analyst’s theory SO = f(v) = av2 + bv + c • I2: There are 6 data points but only three unknowns (a, b, and c)

4. Least Square Error Fit • I3: Now we define a good fit criteria: Least Square Error Square Error (SE) = J(a, b, c)

5. Least Error Fit from Linear Equations • I4: Least Square Error Fit by Solving dJ(a,b,c)/da = 0 (1) dJ(a,b,c)/db = 0 (2) dJ(a,b,c)/dc = 0 (3) Fact: to find min f(x), we solve df(x)/dx = 0 first. Let the solution be x*. If d2f(x*)/dx2 < 0, then x* is the minimum solution. • I5: Equations (1) – (3) are linear equations of a,b, and c.

6. Least Error Fit from Linear Equations • I6: By solving the three linear equations of a, b and c, we find a least square error fit SO = f(v) = av2 + bv + c to the data points. • The square error J(a,b,c) indicates how good the fit is to the 6 data points and we can check how good the analyst’s model is. • This LSE fit method is applicable to • any number of data points • Polynomial function f(.) of any order n and ends up solving a set of linear equations of n-variables (the n coefficients of the order-n polynomial function.

7. Lecture 3: Matrices Today • Properties of Matrix Operations • Inverse • Applications: Economics and Mgmt. and Engrg. Reading Assignment: Secs 2.2-2.5 of Textbook Homework #1 Due http://en.wikipedia.org/wiki/Determinant#Determinants_of_2-by-2_matrices Next Time • Elementary Matrices • Determinant of a Matrix • Evaluation of Determinant Reading Assignment: Secs 3.1-3.2 of Textbook

8. Lecture 3: Matrices Today • Properties of Matrix Operations • Inverse of a Matrix • Applications: Economics and Mgmt. and Engrg.

9. What Did You Actually Learn Last Time?

10. 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

11. 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:

12. Thm 2.1 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

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

14. 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)

15. Transpose of a matrix:

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

17. Properties of transposes:

18. 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

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

20. 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)

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

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

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

24. 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

25. 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

26. 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.

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

28. Thus

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

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

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

32. Power of a square matrix:

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

34. 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

35. 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.

36. 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.

37. Note:

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

39. Applications • Model/Curve Fitting • Failure Prune Machine Capacity Modeling • Network Flow • Power Flow