elementary matrix operations as rank preserving operations

elementary matrix operations as rank preserving operations

Let

M

be a matrix over a

division ring

D

. An

elementary operation

on

M

is any one of the eight operations below:

1.

exchanging two rows

2.

exchanging two columns

3.

adding one row to another

4.

adding one column to another

5.

right multiplying a non-zero scalar to a row

6.

left multiplying a non-zero scalar to a row

7.

right multiplying a non-zero scalar to a column

8.

left multiplying a non-zero scalar to a column

We want to determine the effects of these operations on the various ranks of

M

. To facilitate this discussion, let

M

=

(

a

i

⁢

j

)

be an

n

×

m

matrix and

M

′

=

(

b

i

⁢

j

)

be the matrix after an application of one of the operations above to

M

. In addition, let

v

i

=

(

a

i

⁢

1

,

⋯

,

v

i

⁢

m

)

be the

i

-th row of

M

, and

w

i

=

(

b

i

⁢

1

,

⋯

,

b

i

⁢

m

)

be the

i

-th row of

M

′

. In other words,

M

=

(

v

1

⋮

v

n

)

=

(

a

11

⋯

a

1

⁢

m

⋮

⋱

⋮

a

n

⁢

1

⋯

a

n

⁢

m

)

→

operation

elementary

(

b

11

⋯

b

1

⁢

m

⋮

⋱

⋮

b

n

⁢

1

⋯

b

n

⁢

m

)

=

(

w

1

⋮

w

n

)

=

M

′

Finally, let

d

be the

left row rank

of

M

.

Proposition 1

.

Row and column exchanges preserve all ranks of

M

.

Proof.

Clearly, exchanging two rows of

M

do not change the

subspace

generated by the rows of

M

, and therefore

d

is preserved.

As exchanging rows do not affect

d

, let us assume that rows have been exchanged so that the first

d

rows of

M

are left

linearly independent

.

Now, let

M

′

be obtained from

M

by exchanging columns

i

and

j

. So

w

1

,

…

,

w

n

are vectors obtained respectively from

v

1

,

…

,

v

n

by exchanging the

i

-th and

j

-th

coordinates

. Suppose

r

1

⁢

w

1

+

⋯

+

r

d

⁢

w

d

=

0

. Then we get an

equation

r

1

⁢

b

1

⁢

k

+

⋯

+

r

d

⁢

b

d

⁢

k

=

0

for

1

≤

k

≤

m

. Rearranging these equations, we see that

r

1

⁢

v

1

+

⋯

+

r

d

⁢

v

d

=

0

, which implies

r

1

=

⋯

=

r

d

=

0

, showing that

w

1

,

…

,

w

d

are left linearly independent. This means that

d

is preserved by column exchanges.

Preservation of other ranks of

M

are similarly proved.
∎

Proposition 2

.

Additions of rows and columns preserve all ranks of

M

.

Proof.

Let

M

′

be the matrix obtained from

M

by replacing row

i

by vector

v

i

+

v

j

, and let

V

′

be the left

vector space

spanned by the rows of

M

′

. Since

v

i

+

v

j

∈

V

, we have

V

′

⊆

V

. On other hand,

v

i

=

(

v

i

+

v

j

)

-

v

j

∈

V

′

, so

V

⊆

V

′

, and hence

V

=

V

′

.

Next, let

w

1

,

…

,

w

n

be vectors obtained respectively from

v

1

,

…

,

v

n

such that the

i

-th coordinate of

w

k

is the sum of the

i

-th coordinate of

v

k

and the

j

-th coordinate of

v

k

, with all other coordinates remain the same. Again, by renumbering if necessary, let

v

1

,

…

,

v

d

be left linearly independent. Suppose

r

1

⁢

w

1

+

⋯

+

r

i

⁢

w

i

+

⋯

+

r

d

⁢

w

d

=

0

. A

similar

argument like in the previous proposition shows that

r

1

⁢

v

1

+

⋯

+

(

r

i

+

r

j

)

⁢

v

j

+

r

d

⁢

v

d

=

0

, which implies

r

1

=

⋯

=

r

i

+

r

j

=

⋯

⁢

r

d

=

0

. Since

r

i

=

0

,

r

j

=

0

too. This shows that

w

1

,

…

,

w

d

are left linearly independent, which means that

d

is preserved by additions of columns.

Preservation of other ranks of

M

are proved similarly.
∎

Proposition 3

.

Left (right) non-zero row

scalar multiplication

preserves left (right) row rank of

M

; left (right) non-zero column scalar multiplications preserves left (right) column rank of

M

.

Proof.

Let

w

1

,

…

,

w

n

be vectors obtained respectively from

v

1

,

…

,

v

n

such that the

i

-th vector

w

i

=

r

⁢

v

i

, where

0

≠

r

∈

D

, and all other

w

j

’s are the same as the

v

j

’s. Assume that the first

d

rows of

M

are left linearly independent, and that

i

≤

d

. Suppose

r

1

⁢

w

1

+

⋯

+

r

d

⁢

w

d

=

0

. Then

r

1

⁢

v

1

+

⋯

+

r

i

⁢

(

r

⁢

v

i

)

+

⋯

⁢

r

d

⁢

v

d

=

0

, which implies

r

1

=

⋯

=

r

i

⁢

r

=

⋯

=

r

d

=

0

. Since

r

≠

0

,

r

i

=

0

, and therefore

w

1

,

…

,

w

d

are left linearly independent.

The others are proved similarly.
∎

Proposition 4

.

Left (right) non-zero row scalar multiplication preserves right (left) column rank of

M

; left (right) non-zero column scalar multiplication preserves right (left) row rank of

M

.

Proof.

Let us prove that right multiplying a column by a non-zero scalar

r

preserves the left row rank

d

of

M

. The others follow similarly.

Let

w

1

,

…

,

w

n

be vectors obtained respectively from

v

1

,

…

,

v

n

such that the

i

-th coordinate

b

i

⁢

k

of

w

k

is

a

i

⁢

k

⁢

r

, where

a

i

⁢

k

is the

i

-th coordinate of

v

k

. Suppose once again that the first

d

rows of

M

are left linearly independent, and suppose

r

1

⁢

w

1

+

⋯

+

r

d

⁢

w

d

=

0

. Then for each coordinate

j

we get an equation

r

1

⁢

b

1

⁢

j

+

⋯

+

r

d

⁢

b

d

⁢

j

=

0

. In particular, for the

i

-th coordinate, we have

r

1

⁢

a

1

⁢

j

⁢

r

+

⋯

+

r

d

⁢

a

d

⁢

j

⁢

r

=

0

. Since

r

≠

0

, right multiplying the equation by

r

-

1

gives us

r

1

⁢

a

1

⁢

j

+

⋯

+

r

d

⁢

a

d

⁢

j

=

0

. Re-collecting all the equations, we get

r

1

⁢

v

1

+

⋯

+

r

d

⁢

w

d

=

0

, which implies that

r

1

=

⋯

=

r

d

=

0

, or that

w

1

,

…

,

w

d

are left linearly independent.
∎

Title

elementary matrix operations as rank preserving operations

Canonical name

ElementaryMatrixOperationsAsRankPreservingOperations

Date of creation

2013-03-22 19:22:57

Last modified on

2013-03-22 19:22:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

18

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15-01

Classification

msc 15A33

Classification

msc 15A03

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