elementary matrix

elementary matrix

Elementary Operations on Matrices

Let

𝕄

be the set of all

m

×

n

matrices (over some

commutative ring

R

). An

operation

on

𝕄

is called an

elementary row operation

if it takes a matrix

M

∈

𝕄

, and does one of the following:

1.

interchanges of two rows of

M

,

2.

multiply a row of

M

by a non-zero element of

R

,

3.

add a (

constant

) multiple of a row of

M

to another row of

M

.

An

elementary column operation

is defined similarly. An operation on

𝕄

is an

elementary operation

if it is either an elementary row operation or elementary column operation.

For example, if

M

=

(

a

b

c

d

e

f

)

, then the following operations correspond respectively to the three types of elementary row operations described above

1.

(

a

b

e

f

c

d

)

is obtained by interchanging rows 2 and 3 of

M

,

2.

(

a

b

r

⁢

c

r

⁢

d

e

f

)

is obtained by multiplying

r

≠

0

to the second row of

M

,

3.

(

a

b

c

d

s

⁢

a

+

e

s

⁢

b

+

f

)

is obtained by adding to row 1 multiplied by

s

to row 3 of

M

.

Some immediate observation: elementary operations of type 1 and 3 are always

invertible

. The

inverse

of type 1 elementary operation is itself, as interchanging of rows twice gets you back the original matrix. The inverse of type 3 elementary operation is to add the

negative

of the multiple of the first row to the second row, thus returning the second row back to its original form. Type 2 is invertible provided that the multiplier has an inverse.

Some notation: for each type

k

(where

k

=

1

,

2

,

3

) of elementary operations, let

E

c

k

⁢

(

A

)

be the set of all matrices obtained from

A

via an elementary column operation of type

k

, and

E

r

k

⁢

(

A

)

the set of all matrices obtained from

A

via an elementary row operation of type

k

.

Elementary Matrices

Now, assume

R

has

1

. An

n

×

n

elementary matrix

is a (square) matrix obtained from the

identity matrix

I

n

by performing an elementary operation. As a result, we have three types of elementary matrices, each corresponding to a type of elementary operations:

1.

transposition matrix

T

i

⁢

j

: an matrix obtained from

I

n

with rows

i

and

j

switched,

2.

basic diagonal matrix

D

i

⁢

(

r

)

: a

diagonal matrix

whose entries are

1

except in cell

(

i

,

i

)

, whose entry is a non-zero

element

r

of

R

3.

row replacement matrix

E

i

⁢

j

⁢

(

s

)

:

I

n

+

s

⁢

U

i

⁢

j

, where

s

∈

R

and

U

i

⁢

j

is a

matrix unit

with

i

≠

j

.

For example, among the

3

×

3

matrices, we have

T

12

=

(

0

1

0

1

0

0

0

0

1

)

,

D

3

⁢

(

r

)

=

(

1

0

0

0

1

0

0

0

r

)

,

and

E

32

⁢

(

s

)

=

(

1

0

0

0

1

0

0

s

1

)

For each positive integer

n

, let

𝔼

k

⁢

(

n

)

be the

collection

of all

n

×

n

elementary matrices of type

k

, where

k

=

1

,

2

,

3

.

Below are some basic

properties

of elementary matrices:

•

T

i

⁢

j

=

T

j

⁢

i

, and

T

i

⁢

j

2

=

I

n

.

•

D

i

⁢

(

r

)

⁢

D

i

⁢

(

r

-

1

)

=

I

n

, provided that

r

-

1

exists.

•

E

i

⁢

j

⁢

(

s

)

⁢

E

i

⁢

j

⁢

(

-

s

)

=

I

n

.

•

det

⁡

(

T

i

⁢

j

)

=

-

1

,

det

⁡

(

D

i

⁢

(

r

)

)

=

r

, and

det

⁡

(

E

i

⁢

j

⁢

(

s

)

)

=

1

.

•

If

A

is an

m

×

n

matrix, then

E

c

k

⁢

(

A

)

=

{

A

⁢

E

∣

E

∈

𝔼

k

⁢

(

n

)

}

and

E

r

k

⁢

(

A

)

=

{

E

⁢

A

∣

E

∈

𝔼

k

⁢

(

m

)

}

.

•

Every non-singular matrix can be written as a

product

of elementary matrices. This is the same as saying: given a non-singular matrix, one can perform a finite number of

elementary

row (column) operations on it to obtain the identity matrix.

Remarks

.

•

One can also define elementary matrix operations on matrices over general rings. However, care must be taken to consider left

scalar multiplications

and right scalar multiplications as separate operations.

•

The discussion above pertains to elementary

linear algebra

. In algebraic K-theory, an elementary matrix is defined only as a row replacement matrix (type 3) above.

Title

elementary matrix

Canonical

name

ElementaryMatrix

Date of creation

2013-03-22 18:30:38

Last modified on

2013-03-22 18:30:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15-01

Related topic

MatrixUnit

Related topic

GaussianElimination

Defines

elementary operation

Defines

elementary column operation

Defines

elementary row operation

Defines

basic diagonal matrix

Defines

transposition matrix

Defines

row replacement matrix

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