rank of a matrix

rank of a matrix

Let

D

be a

division ring

, and

M

an

m

×

n

matrix over

D

. There are four numbers we can

associate

with

M

:

1.

the

dimension

of the

subspace

spanned by the columns of

M

viewed as elements of the

n

-dimensional right

vector space

over

D

.

2.

the dimension of the subspace spanned by the columns of

M

viewed as elements of the

n

-dimensional left vector space over

D

.

3.

the dimension of the subspace spanned by the rows of

M

viewed as elements of the

m

-dimensional right vector space over

D

.

4.

the dimension of the subspace spanned by the rows of

M

viewed as elements of the

m

-dimensional left vector space over

D

.

The numbers are respectively called the

right column rank

,

left column rank

,

right row rank

, and

left row rank

of

M

, and they are respectively denoted by

rc

.

rnk

⁡

(

M

)

,

lc

.

rnk

⁡

(

M

)

,

rr

.

rnk

⁡

(

M

)

, and

lr

.

rnk

⁡

(

M

)

.

Since the columns of

M

are the rows of its

transpose

M

T

, we have

lc

.

rnk

⁡

(

M

)

=

lr

.

rnk

⁡

(

M

T

)

,

and

rc

.

rnk

⁡

(

M

)

=

rr

.

rnk

⁡

(

M

T

)

.

In

addition

, it can be shown that for a given matrix

M

,

lc

.

rnk

⁡

(

M

)

=

rr

.

rnk

⁡

(

M

)

,

and

rc

.

rnk

⁡

(

M

)

=

lr

.

rnk

⁡

(

M

)

.

For any

0

≠

r

∈

D

, it is also easy to see that the left column and row ranks of

r

⁢

M

are the same as those of

M

. Similarly, the right column and row ranks of

M

⁢

r

are the same as those of

M

.

If

D

is a field,

lc

.

rnk

⁡

(

M

)

=

rc

.

rnk

⁡

(

M

)

, so that all four numbers are the same, and we simply call this number the

rank

of

M

, denoted by

rank

⁡

(

M

)

.

Rank can also be defined for matrices

M

(over a fixed

D

) that satisfy the

identity

M

=

r

⁢

M

T

, where

r

is in the center of

D

. Matrices satisfying the identity include

symmetric

and

anti-symmetric

matrices.

However, the left column rank is not necessarily the same as the right row rank of a matrix, if the underlying division ring is not

commutative

, as can be shown in the following example: let

u

=

(

1

,

j

)

and

v

=

(

i

,

k

)

be vectors over the

Hamiltonian quaternions

ℍ

. They are columns in the

2

×

2

matrix

M

:=

(

1

i

j

k

)

Since

i

⁢

u

=

(

i

,

i

⁢

j

)

=

(

i

,

k

)

=

v

, they are left

linearly dependent

, and therefore the left column rank of

M

is

1

. Now, suppose

u

⁢

r

+

v

⁢

s

=

(

0

,

0

)

, with

r

,

s

∈

ℍ

. Since

u

⁢

i

=

(

i

,

j

⁢

i

)

=

(

i

,

-

k

)

, then

u

⁢

i

⁢

(

-

i

⁢

r

)

+

v

⁢

s

=

0

, which boils down to two

equations

i

⁢

r

=

s

and

-

i

⁢

r

=

s

, and which imply that

s

=

r

=

0

, showing that

u

,

v

are right

linearly independent

. Thus the right column rank of

M

is

2

.

Title

rank of a matrix

Canonical

name

RankOfAMatrix

Date of creation

2013-03-22 19:22:42

Last modified on

2013-03-22 19:22:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A03

Classification

msc 15A33

Related topic

DeterminingRankOfMatrix

Defines

left row rank

Defines

left column rank

Defines

right row rank

Defines

right column rank

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