More on Inverse

1. More on Inverse

2. Last Week Review • Matrix • Rule of addition • Rule of multiplication • Transpose • Main Diagonal • Dot Product • Block Multiplication • Matrix and Linear Equations • Basic Solution • X1 + X0 • Linear Combination • All solutions of LES • Inverse • Det • Matrix Inversion Method • Double matrix

3. Warm Up • Find the inverse of • Using matrix inversion method • [ A I ]  [ I A-1 ]

4. Solution • Start with the double matrix

5. Swap 1 with 2 • R2 – 2R1, R3 – R1

6. More to Reduced Row Echelon Form

7. Properties of Inverse

8. Transpose and Inverse • If A is invertible, show that • AT is also invertible • (AT)-1 = (A-1)T

9. Solution • A-1 exists • Its transpose is the inverse of AT • So AT(A-1)T = (A-1A)T = IT = I (A-1)TAT = (AA-1)T = IT = I

10. Inverse of Multiplication • If A and B are invertible, show that • AB is also invertible • (AB)-1 = B-1A-1

11. Solution • Assume that (AB)-1 exists • And it is B-1A-1 • (B-1A-1)(AB) = B-1(A-1A)B = B-1IB = B-1B = I • (AB)(B-1A-1) = A(BB-1)A-1 = AIA-1 = AA-1 = I • Hence, it is actually the inverse

12. Rule of Inverse

13. Inverse Equivalence • A is invertible • The homogeneous system AX = 0 has only the trivial solution X = 0 • A can be carried to the identity matrix In by elementary row operation • The system AX=B has at least one solution X for every choice of B • There exists an n x n matrix C such that AC = In

14. ELEMENTARY matrices

15. Elementary Matrix • A matrix that can be obtained from I by single elementary row operation • Example

16. Elementary Operation • Interchange two equations • Multiply one equation with a nonzero number • Add a multiple of one equation to a different equation

17. Lemma • If an elementary row operation is performed on an m x n matrix A • The result is EA where E is the elementary matrix • E is obtained by performing the same operation on m x m identity matrix.

18. Inverse of elementary operation • Each operation has an inverse • Also an elementary operation • So are the elementary matrix

19. Inverse of Elementary Matrix • Hence, each elementary matrix E has its inverse • The inverse change E back to I

20. Lemma 2 • Every elementary matrix E is invertible • Its inverse is also an elementary matrix • Of the same type as well • It also corresponds to the inverse of the row operation that produce E

21. Inverse and Rank • Suppose that A  B by a series of elementary row operation • Hence • A  E1A  E2E1A  EkEk-1…E2E1A  B • i.e., A  UA = B • Where U = EkEk-1…E2E1 • U is invertible • Why?

22. Finding U • AB by some elementary row operations • Perform the same operations on I • Doing the same thing just like the matrix inversion algorithm • [A I]  [B U]

23. Theorem: Property of U • Suppose that A is m x n and A  B by some sequence of elementary row operations • B = UA where U is m x m invertible matrix • U can be computed by [A I]  [B U] using the same operations • U = EkEk-1…E2E1 where each Ei is the elementary matrix corresponding to the elementary row operation

24. U and A-1 • Suppose that A is invertible • We know that A  I • So, let B be I • Hence, [A I]  [I U] • I = UA • i.e., U = A-1 • This is exactly the matrix inversion algorithm • But, A-1 =U = EkEk-1…E2E1 • Hence A = (A-1)-1 = (EkEk-1…E2E1)-1= E1-1E2-1…Ek-1-1Ek-1 • This means that every invertible matrix is a product of elementary matrices!!!

25. Theorem 2 • A square matrix is invertible if and only if it is a product of elementary matrices.

26. Transformation

27. Ordered n-tuple (Vector) • Let R be the set of real number • If n >= 1, an ordered sequence • (a1,a2,..,an) is called an ordered n-tuple • RN denotes the set of all ordered n-tuples • The ordered n-tuple is also called vectors

28. Transformation • A function T from RN toRM • Written T: RN RM • RN domain • RM codomain • To describe T, we must give the definition of all T(X) for every X in RN • T and S is the same if T(X) = S(X) for every X • That is the definition of T

29. Matrix Transformation • A transformation such that • T(X) is AX • Called the matrix transformation induced by A • If A = 0, it is called the zero transformation • If A = I, it is called the identify transformation

30. Example • X-expansion • Induced by

31. Example • Reflection • Induced by

32. Example • X-shear • Induced by

33. Translation is not Linear Transform • Translation • T(X) = X + w • If it is, then • X + w = AX for some A • What if a = 0?

34. Linear Transformation • A transformation is called a linear transformation when • T(X + Y) = T(X) + T(Y) • T(aX) = aT(X)

35. Linear Transform and Matrix Transform • Let T:RN RM be a transformation • T is linear if and only if it is a matrix transformation • If T is linear, then T is induced by a unique matrix A

36. Composition • Transform of a transform • ST = S(T(X))

37. Composition • If R,S,T are linear transformation • Compositions of them are also linear • Is associative • (since it is matrix transform)

38. Inverse through transform • Inverse of the transform is the inverse of the function • Hence, domain and codomain must be the same • Given a linear transformation • It’s inverse is induced by A-1