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

• X1 + X0

– 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– AT is also invertible– (AT)-1 = (A-1)T

Solution

• A-1 exists– Its transpose is the inverse of AT

• SoAT(A-1)T = (A-1A)T = IT = I

(A-1)TAT = (AA-1)T = IT = I

Inverse of Multiplication

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

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

Rule of Inverse

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

ELEMENTARY MATRICES

Elementary Matrix

• A matrix that can be obtained from I by single elementary row operation

• Example

Elementary Operation

• Interchange two equations• Multiply one equation with a nonzero

number• Add a multiple of one equation to a

different equation

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.

Inverse of elementary operation

• Each operation has an inverse– Also an elementary operation

• So are the elementary matrix

Operation Inverse

Interchange row p and q Interchange row q and p

Multiply row p by k != 0 Multiply row p by 1/k

Add k times row p to row q != p Subtract 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 E1A E2E1A EkEk-1…E2E1A B

• i.e., A UA = B– Where U = EkEk-1…E2E1

• U is invertible– Why?

Finding U

• AB 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 = EkEk-1…E2E1 where each Ei 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 = 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!!!

Theorem 2

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

TRANSFORMATION

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

Transformation

• A function T from RN to RM

• 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

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

• 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

Composition

• Transform of a transform

• ST = 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