More on Inverse

Last Week Review Matrix – Rule of addition – Rule of multiplication – Transpose – Main Diagonal – Dot Product Block Multiplication Matrix and Linear Equations – Basic Solution X 1 + X 0 – Linear Combination – All solutions of LES Inverse – Det – Matrix Inversion Method Double matrix

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

Solution Start with the double matrix

Swap 1 with 2 R2 – 2R1, R3 – R1

More to Reduced Row Echelon Form

PROPERTIES OF INVERSE

Transpose and Inverse If A is invertible, show that – A T is also invertible – (A T ) -1 = (A -1 ) T

Solution A -1 exists – Its transpose is the inverse of A T So A T (A -1 ) T = (A -1 A) T = I T = I (A -1 ) T A T = (AA -1 ) T = I T = I

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

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

Rule of Inverse

Inverse Equivalence

ELEMENTARY MATRICES

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

Elementary Operation

Lemma

Inverse of elementary operation Each operation has an inverse – Also an elementary operation So are the elementary matrix OperationInverse Interchange row p and qInterchange row q and p Multiply row p by k != 0Multiply row p by 1/k Add k times row p to row q != pSubtract k times row p to row q

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

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

Inverse and Rank Suppose that A  B by a series of elementary row operation Hence – A  E 1 A  E 2 E 1 A  E k E k-1 …E 2 E 1 A  B i.e., A  UA = B – Where U = E k E k-1 …E 2 E 1 U is invertible – Why?

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]

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 = E k E k-1 …E 2 E 1 where each E i is the elementary matrix corresponding to the elementary row operation

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 = E k E k-1 …E 2 E 1 – Hence A = (A -1 ) -1 = (E k E k-1 …E 2 E 1 ) -1 = E 1 -1 E 2 -1 …E k-1 -1 E k -1 This means that every invertible matrix is a product of elementary matrices!!!

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

TRANSFORMATION

Ordered n-tuple (Vector)

Transformation

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

Example X-expansion Induced by

Example Reflection Induced by

Example X-shear Induced by

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

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

Linear Transform and Matrix Transform

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

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

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