1. ENGG2013 Unit 7 Non-singular matrix and Gauss-Jordan elimination Jan, 2011.

2. Outline Matrix arithmetic – Matrix addition, multiplication Non-singular matrix Gauss-Jordan elimination kshumENGG20132

3. The love function: a normal case kshumENGG20133 Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E Function L Domain Range Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E Domain Range L(Boy 1) = Girl A, but L’(Girl A) = Boy 4. Function L’

4. The love function: a utopian case kshumENGG20134 Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E Function L Domain Range Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E Function L’ Domain Range This function L’ is the inverse of L

5. The love function: no inverse kshumENGG20135 Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E Function L Domain Range Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E Domain Range This function L has no inverse This is not a function

6. Undo-able kshumENGG20136 Multiplied by Rotate 90 degrees clockwise Multiplied by Rotate 90 degrees counter-clockwise A matrix which represents a reversible process is called invertible or non-singular.

7. Objectives How to determine whether a matrix is invertible? If a matrix is invertible, how to find the corresponding inverse matrix? kshumENGG20137

8. MATRIX ALGEBRA kshumENGG20138

9. Matrix equality Two matrices are said to be equal if 1.They have the same number of rows and the same number of columns (i.e. same size). 2.The corresponding entry are identical. kshumENGG20139

10. Matrix addition and scalar multiplication We can add two matrices if they have the same size To multiply a matrix by a real number, we just multiply all entries in the matrix by that number. kshumENGG201310

11. Matrix multiplication Given an m  n matrix A and a p  q matrix B, their product AB is defined if n=p. If n = p, we define their product, say C = AB, by computing the (i,j)-entry in C as the dot product of the i-th row of A and the j-th row of B. kshumENGG201311 mnmn pqpq m  q

12. Examples kshumENGG201312 is undefined.

13. Square matrix A matrix with equal number of columns and rows is called a square matrix. For square matrices of the same size, we can freely multiply them without worrying whether the product is well-defined or not. – Because multiplication is always well-defined in this case. The entries with the same column and row index are called the diagonal entries. – For example: kshumENGG201313

14. Compatibility with function composition kshumENGG201314 Multiplied by

15. Order does matter in multiplication kshumENGG201315 Multiplied by Rotate 90 degrees Multiplied by Reflection around x-axis Multiplied by Reflection around x-axis Multiplied by Rotate 90 degrees Are they the same?

16. Non-commutativity For real numbers, we have 3  5 = 5  3. – Multiplication of real numbers is commutative. For matrices, in general AB  BA. – Multiplication of matrices is non-commutative. – For example kshumENGG201316

17. Associativity For real numbers, we have (3  4)  5 = 3  (4  5). – Multiplication of real numbers is associative. For any three matrices A, B, C, it is always true that (AB)C = A(BC), provided that the multiplications are well-defined. – Multiplication of matrices is associative. kshumENGG201317

18. INVERTIBLE MATRIX kshumENGG201318

19. Identity matrix A square matrix whose diagonal entries are all one, and off-diagonal entries are all zero, is called an identity matrix. We usually use capital letter I for identity matrix, or add a subscript and write I n if we want to stress that the size is n  n. kshumENGG201319

20. Multiplication by identity matrix Identity matrix is like a do-nothing process. – There is no change after multiplication by the identity matrix IA = A for any A. BI = B for any B. kshumENGG201320 Multiplied by is trivial

21. Invertible matrix Given an n  n matrix A, if we can find a matrix A’, such that then A is said to be invertible, or non-singular. This matrix A’ is called an inverse of A. kshumENGG201321 Multiplied by A Multiplied by A’ Multiplied by I n

22. Example kshumENGG201322 Multiplied by Rotate 90  CW Multiplied by Rotate 90  CCW implies is invertible.

23. Matrix inverse may not exist If matrix A induces a many-to-one mapping, then we cannot hope for any inverse. kshumENGG201323 has no inverse

24. Naïve method for computing matrix inverse Consider Want to find A’ such that A A’= I Solve for p, q, r, s in kshumENGG201324

25. Uniqueness of matrix inverse Before we discuss how to compute matrix inverse, we first show there is at most one A’ such that A A’ = A’ A = I. Suppose on the contrary that there is another matrix A’’ such that A A’’ = A’’ A = I. We want to prove that A’ = A’’. kshumENGG201325

26. Proof of uniqueness kshumENGG201326 Defining property of A’’ Multiply by A’ from the left I times anything is the same thing Matrix multiplication is associative Defining property of A’ I times anything is the same thing

27. Notation Since the matrix inverse (if exists) is unique, we use the symbol A -1 to represent the unique matrix which satisfies We say that A -1 is the inverse of A. kshumENGG201327

28. A convenient fact To check that a matrix B is the inverse of A, it is sufficient to check either 1.BA = I, or 2.AB = I. It can be proved that (1) implies (2), and (2) implies (1). – The details is left as exercise. kshumENGG201328

29. GAUSS-JORDAN ELIMINATION kshumENGG201329

30. Row operation using matrix Recall that there are three kind of elementary row operations 1.Row exchange 2.Multiply a row by a non-zero constant 3.Replace a row by the sum of itself and a constant multiple of another row. We can perform elementary row operation by matrix multiplication (from the left). All three kinds of operation are invertible. kshumENGG201330

31. Row exchange Example: exchange row 2 and row 3 Multiply the same matrix from the left again, we get back the original matrix. kshumENGG201331

32. Multiply a row by a constant Multiply the first row by -1. Multiply the same matrix from the left again, we get back the original matrix. kshumENGG201332

33. Row replacement Add the first row to the second row Multiply by another matrix from the left to undo kshumENGG201333

34. Elementary matrix (I) Three types of elementary matrices 1.Exchange row i and row j kshumENGG201334 Row i Row j Col. jCol. i

35. Elementary matrix (II) 2.Multiply row i by m kshumENGG201335 Row i Col. i

36. Elementary matrix (III) 3.Add s times row i to row j kshum ENGG2013 36 Row i Row j Col. jCol. i

37. Row reduction A series of row reductions is the same as multiplying from the left a series of elementary matrices. kshumENGG201337 … E 1, E 2, E 3, … are elementary matrices.

38. If we can row reduce to identity Then A is non-singular, or invertible. kshumENGG201338 (Matrix multiplication is associative)

39. Gauss-Jordan elimination It is convenient to append an identity matrix to the right We can interpret it as If we can row reduce A to the identity by a series of row operations then we can apply the same series of row operations to I and obtain the inverse of A. kshumENGG201339

40. Algorithm Input: an n  n matrix A. Create an n  2n matrix M – The left half is A – The right half is I n Try to reduce the expanded matrix M such that the left half is equal to I n. If succeed, the right half of M is the inverse of A. If you cannot reduce the left half of M to, then A is not invertible, a.k.a. singular. kshumENGG201340

41. Example Find the inverse of 1.Create a 3  6 matrix 2.After some row reductions we get Answer: kshumENGG201341