Matrices Dr.s.s.chauhan (1)

8/10/2019 Matrices Dr.s.s.chauhan (1)

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MATRICESCompiled by Dr. S.S. Chauhan


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An m x n matrix is an array of mxn objectsin m rows and n columns and is representedin the form

Definition

][

321

3333231

2232221

1131211

nmmnmmm

n

n

n

ij

aaaa

aaaa

aaaa

aaaa

a A


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3

Squarematrix:

If m = n , then the matrix is a square

matrixElements a ij for i=j called diagonalelements and

is called the trace of A.

Types of Matrices

...11 221

n

ii nn

i

a a a a


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Partitioned matrices:

2221

1211

34333231

24232221

14131211

A A

A A

aaaa

aaaaaaaa

A

submatrix

3

2

1

34333231

24232221

14131211

r

r r

aaaa

aaaaaaaa

A

432134333231

24232221

14131211

cccc

aaaa

aaaaaaaa

A


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Then (1) A + B = B + A

(2) A + ( B + C ) = ( A + B ) + C

(3) ( cd ) A = c ( dA )

(4) 1 A = A

(5) c ( A + B ) = c A + c B

(6) ( c + d ) A = cA + dA

scalar :, ,,, If d c M C B A nm

Properties of matrix addition and scalar multiplication:


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calar c M A nm s: , If

A A nm 0(1)Then

nm A A 0)((2)

nmnm or AccA 000)3(

Notes:

(1) 0 m n : the additive identity for the set of all m n matrices

(2) A : the additive inverse of A

Properties of zero matrices:


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A A T T )( )1(T T T B A B A )( )2(

)()( )3( T T AccA

)( )4( T T T A B AB

Properties of transposes:


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1nEliminatioJordan-Gauss || A I I A Ex 2: (Find the inverse of the matrix)

31

41 A

Sol:

I AX

10

01

31

41

2221

1211

x x

x x

10

01

33

44

22122111

22122111

x x x x

x x x x

Find the inverse of a matrix by Gauss-Jordan Elimination:


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110

301

031

141)1()4(

21)1(

12 , r r

110401

131041)2(

)4(21

)1(12 , r r

1,3 2111 x x

1,4 2212 x x

)( 11

43 1-2221

12111 AA I AX x x

x x A X

Thus

(2) 1304

(1) 0314

2212

2212

2111

2111

x x x x

x x x x


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Rank of a Matrix

A number r is said to be rank of amatrix A, if there existsa non zero minor of order r and all

minors oforder r+1 vanish.

Or equivalentlyThe maximum number of linearlyindependent rows.


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I A 0

(1)

0)( )2(factors

k A AA Ak

k

integers:, )3( sr A A A sr sr

rs sr A A )(

k n

k

k

k

n d

d

d

D

d

d

d

D

00

00

00

00

00

00

)4(2

1

2

1

Power of a square matrix:


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Thm 1: (Systems of equations with unique solutions)

If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by

b A x 1

Pf:

( A is nonsingular)

b A x

b A Ix

b A Ax Ab Ax

1

1

11

This solution is unique.

.equationof solutions twowereandIf 21 b Ax x x

21then Axb Ax 21 x x (Left cancellation property)


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2.4 Elementary MatricesRow elementary matrix:

An n n matrix is called an elementary matrix if it can be obtained

from the identity matrix I n by a single elementary operation.

Three row elementary matrices:

)()1( I r R ijij )0( )()2( )()( k I r R k i

k i

)()3( )()( I r R k ijk

ij

Interchange two rows.Multiply a row by a nonzeroconstant.Add a multiple of a row toanother row.

Note:Only do a single elementary row operation.


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Notes:

A R Ar ijij )( )1(

A R Ar k ik

i)()( )( )2(

A R Ar k ijk

ij)()( )( )3(

EA Ar

E I r

)(

)(

Thm 2.12: (Representing elementary row operations)Let E be the elementary matrix obtained by performing anelementary row operation on I m . If that same elementary rowoperation is performed on an m n matrix A, then the resultingmatrix is given by the product EA .


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If A is an n n matrix, then the following statements are equivalent.

(1) A is invertible.

(2) A x = b has a unique solution for every n 1 column matrix b.

(3) A x = 0 has only the trivial solution.

(4) A is row-equivalent to I n .

(5) A can be written as the product of elementary matrices.

Thm 2.15: (Equivalent conditions)


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LU A L is a lower triangular matrix

U is an upper triangular matrix

If the n n matrix A can be written as the product of a lowertriangular matrix L and an upper triangular matrix U , thenA=LU is an LU-factorization of A

Note:

If a square matrix A can be row reduced to an upper triangularmatrix U using only the row operation of adding a multiple ofone row to another row below it, then it is easy to find an LU -factorization of A .

LU A

U E E E A

U A E E E

k

k

112

11

12

LU -factorization:


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Eigen Values and Eigen Vectors

Definition. A nonzero vector x is an eigenvector of asquare matrix A if there exists a scalar such that Ax = x . Then is an eigenvalue of A.

Note : The zero vector can not be an eigenvector eventhough A0 = 0 . But = 0 can be an eigenvalue.


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Properties of eigenvalues and eigenvectors Property 1: The sum of the eigenvalues ofa matrix equals the trace of the matrix.

Property 2: A matrix is singular if and onlyif it has a zero eigenvalue.

Property 3: The eigenvalues of an upper(or lower) triangular matrix are theelements on the main diagonal.

Property 4: If is an eigenvalue of A andA is invertible, then 1/ is an eigenvalueof matrix A -1 .


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Property 5: If is an eigenvalue of A then k isan eigenvalue of kA where k is any arbitrary

scalar.Property 6: If is an eigenvalue of A then k isan eigenvalue of Ak for any positive integer k .

Property 7: If is an eigenvalue of A then isan eigenvalue of AT.

Property 8: The product of the eigenvalues(counting multiplicity) of a matrix equals the

determinant of the matrix.


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Matrices Dr.s.s.chauhan (1)