More on Inverse - PowerPoint Presentation

Slide1

More on InverseSlide2

Last Week Review

Matrix

Rule of addition

Rule of multiplicationTransposeMain DiagonalDot ProductBlock Multiplication

Matrix and Linear Equations

Basic Solution

X

1

+ X

0

Linear Combination

All solutions of LES

Inverse

Det

Matrix Inversion Method

Double matrixSlide3

Warm Up

Find the inverse of

Using matrix inversion method

[ A I ]  [ I A-1

]Slide4

Solution

Start with the double matrixSlide5

Swap 1 with 2

R2 – 2R1, R3 – R1Slide6

More to Reduced Row Echelon FormSlide7

Properties of InverseSlide8

Transpose and Inverse

If A is invertible, show that

A

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

-1

)

TSlide9

Solution

A

-1

existsIts transpose is the inverse of ATSo

A

T

(A

-1

)

T

=

(A

-1

A)

T

= I

T

=

I

(A

-1

)

T

A

T

= (AA

-1

)

T

= I

T

= ISlide10

Inverse of Multiplication

If A and B are invertible, show that

AB is also invertible

(AB)-1 = B-1A-1Slide11

Solution

Assume that (AB)

-1

existsAnd it is B-1A-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 inverseSlide12

Rule of InverseSlide13

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 operationThe 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 = I

nSlide14

ELEMENTARY matricesSlide15

Elementary Matrix

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

ExampleSlide16

Elementary Operation

Interchange two equations

Multiply one equation with a

nonzero

number

Add a multiple of one equation to a

different

equationSlide17

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.Slide18

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 Slide19

Inverse of Elementary Matrix

Hence, each elementary matrix E has its inverse

The inverse change E back to ISlide20

Lemma 2

Every elementary matrix E is invertible

Its inverse is also an elementary matrix

Of the same type as wellIt also corresponds to the inverse of the row operation that produce ESlide21

Inverse and Rank

Suppose that A

 B by a series of elementary row operation

HenceA  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?Slide22

Finding U

A

B by some elementary row operations

Perform the same operations on IDoing the same thing just like the matrix inversion algorithm

[A I]  [B U]Slide23

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 matrixU 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 operationSlide24

U and A-1

Suppose that A is invertible

We know that A

 ISo, 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!!!Slide25

Theorem 2

A square matrix is invertible if and only if it is a product of elementary matrices.Slide26

TransformationSlide27

Ordered n-tuple

(Vector)

Let

R

be the set of real number

If n >= 1, an ordered sequence

(a

1

,a

2

,..,a

n

) is called an ordered n-

tuple

R

N

denotes the set of all ordered n-

tuples

The ordered n-

tuple

is also called vectorsSlide28

Transformation

A function T from

R

N

to

R

M

Written T:

R

N



R

M

R

N

domain

R

M

codomain

To describe T, we must give the definition of all T(X) for every X in

R

N

T and S is the same if T(X) = S(X) for every X

That is the definition of TSlide29

Matrix Transformation

A transformation such that

T(X) is AX

Called the matrix transformation induced by AIf A = 0, it is called the zero transformationIf A = I, it is called the identify transformationSlide30

Example

X-expansion

Induced bySlide31

Example

Reflection

Induced bySlide32

Example

X-shear

Induced bySlide33

Translation is not Linear Transform

Translation

T(X) = X + w

If it is, thenX + w = AX for some AWhat if a = 0?Slide34

Linear Transformation

A transformation is called a linear transformation when

T(X + Y) = T(X) + T(Y)

T(aX) = aT(X)Slide35

Linear Transform and Matrix Transform

Let T:

R

N



R

M

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 ASlide36

Composition

Transform of a transform

S

T = S(T(X))Slide37

Composition

If R,S,T are linear transformation

Compositions of them are also linear

Is associative(since it is matrix transform)Slide38

Inverse through transform

Inverse of the transform is the inverse of the function

Hence, domain and

codomain must be the sameGiven a linear transformationIt’s inverse is induced by A-1