Elementary Operations of MatrixElementary operations of matrix are an important tools in linear algebra.

The next three operations are called as No.1st,2ndand3rd elementary transforms of matrix.

)(or )( :follows as

and ns)rows(colum einterchang we)(

jijiijij ccrrcrj

ii

).(:follows as

kby )row(column hemultiply t we)(

ii kckr

iii

).(:)asrow(columnith the

to)row(columnjth thek times add we)(

jiji kcckrr

iii

4131

122122

2832

A

2832

122122

4131

31 rr

6690

4460

41311312

22

rrrr

2230

2230

4131

0000

2230

4131

.BAor BA asmark we

operations elemantaryby

B intoA ormcan transf weIf

，

How to transform a matrix into an echelon oneFunction

Elementary operations of matrix about row and column are all called elementary operations

.

301

020

201

A

500

020

201

9113

12334

3221

B

0770

011110

3221

0000

0110

3221

100

010

001

0000

0010

0001

Equivalent Matrix If matrix A can be transformed into B by some elementary operations, we say A and B are equivalent,and we marked as A B.

Equivalent Matrices have the following characters：CACBBAABBAAA ,;;

nmIA

00000

00000

00100

00010

00001

normal form of A

Theory: any matrix has its own

normal form.

As last e.g.：

0000

2230

4131

A

0000

2230

0001

1413

12

4

3

cccc

cc

0000

0010

0001

32

32

3

2

24

23

r

cc

cc

Corollary:matrices A and B are equivalent if and only if they have the same normal forms.

Rank of Matrix

A. ofnant subdetermik order an

called is scalars ofarray resulting theout, picked are

columnsk and rowsk matrix.Ifn man beA Let .1

.in nantssubdetermiorder - are There nmkn

km AkCC

2.The highest order of nonzero subdeterminantsof A is called the rank of A and denoted by r(A).

Obviously:r(O)=0.As long as A is not 0, r(A)>0. And :};,min{)()( nmAri nm

.)(

zero, are nantssubdetermiorder -r all if;)(

nant,subdetermiorder -r nonzero a is thereIf)(

rAr

rAr

ii

).()()( ArAriii T e.g.: Compute the rank of Matrix A.

00

00

2222

111211

rnrr

nr

nr

aa

aaa

aaaa

A

.)(Obviously rAr

We can figure out the rank of the matrix by elementary operations

)0( 2211 rraaa

The Rank:Theory: Elementary operations do not change the resulting scalar of the rank.

Prove: only after elementary row operations.

);()( BrArBAijr

);()( BrArBAikr

mnmm

jnjj

inii

n

aaa

aaa

aaa

aaa

A

21

21

21

11211

ji krr

B

aaa

aaa

kaakaakaa

aaa

mnmm

jnjj

jninjiji

n

21

21

2211

11211

We can conclude:

)()()()( BrArBrAr )()( BrAr

e.g. To figure out the rank of matrices

4131

122122

2832

.1 A

2832

122122

4131

31 rrA

6690

4460

4131

0000

4460

41312)( Ar

9300

121070

2220

4321

9300

5300

1110

4321

9300

121070

1110

4321

4000

5300

1110

4321

4)( Br

5021

0113

2101

4321

.2 B

9113

1234

3221

tA ？， 3)(? ifonly Art

0770

01180

3221

tA

01180

0110

3221

t

0300

0110

3221

t

.3)(,3 Art

Elementary Matrices

Definition: An elementary matrix is one obtained by performing a single elementary operationon an identity matrix.The followings are three different kinds of elementary matrices:

1

01

10

1

),(

jiE

They also include the ones obtained by performing a single elementary column operation on an identity matrix.

1

1

))((

k

kiE

1

1

1

1

))(,(

k

kjiE

Properties of Elementary Matrices1.

),(),( jiEjiET

1

01

10

1

),(

TjiE ),( jiE

))((

1

1

))((

kiE

kkiE T

))(,(

1

1

1

1

))(,( kijE

k

kjiE T

))(())(( kiEkiET

))(,())(,( kijEkjiET

Transposes of elementary matrices are sameto themselves.

1),( jiE2. kkiE ))(( 1))(,( kjiE

Elementary matrices are all invertible.

The Relation Between Elementary Matrices and Elementary Operations.

Look at an example first.

333231

232221

131211

aaa

aaa

aaa

A

100

001

010

)2,1(E

333231

232221

131211

100

001

010

)2,1(

aaa

aaa

aaa

AE

333231

131211

232221

aaa

aaa

aaa

Elementary row operations are equivalent to multiply anelementary matrix on the left. And multiplying an elementary matrix on the right stands for Elementary column operations.

nmmmrownmrow AETAT )()(

)()( nncolumnnmnmcolumn ETAAT

100

001

010

)2,1(

333231

232221

131211

aaa

aaa

aaa

AE

333132

232122

131112

aaa

aaa

aaa

e.g. To find the normal form of matrix A and useelementary matrices to show the elementary operations

011

110

001

A

010

110

001

13 rr

110

010

001

32 rr

100

010

001

23 rrI

101

010

001

1P

010

100

001

2P

110

010

001

3PIAPPP 123

We can prove it

333231

232221

131211

aaa

aaa

aaa

A

133312321131

131211

232221

aaaaaa

aaa

aaa

B

100

001

010

1P

101

010

001

2P

BPAP 21)1(BPAP 12)2(

BAPP 21)3(BAPP 12)4(

4131

122122

2832

101

010

001

1000

0120

0010

0001

6903

122122

2832

1000

0120

0010

0001

69183

122162

28192