Smith normal form

Smith normal form

This topic gives a version of the Gauss elimination algorithm for a

commutative

principal ideal domain

which is usually described only for a field.

Let

A

≠

0

be a

m

×

n

-matrix with entries from a commutative principal ideal
domain

R

. For

a

∈

R

∖

{

0

}

δ

⁢

(

a

)

denotes the number of

prime factors

of

a

. Start with

t

=

1

and choose

j

t

to be the smallest
column index of

A

with a non-zero entry.

(I)

If

a

t

,

j

t

=

0

and

a

k

,

j

t

≠

0

, exchange rows 1 and

k

.

(II)

If there is an entry at position

(

k

,

j

t

)

such that

a

t

,

j

t

∤

a

k

,

j

t

, then set

β

=

gcd

⁡

(

a

t

,

j

t

,

a

k

,

j

t

)

and choose

σ

,

τ

∈

R

such that

σ

⋅

a

t

,

j

t

-

τ

⋅

a

k

,

j

t

=

β

.

By left-multiplication with an appropriate matrix it can be achieved that row 1
of the

matrix product

is the sum of row 1 multiplied by

σ

and row

k

multiplied by

(

-

τ

)

. Then we get

β

at position

(

t

,

j

t

)

, where

δ

⁢

(

β

)

<

δ

⁢

(

a

t

,

j

t

)

.
Repeating these steps one obtains a matrix having an entry at position

(

t

,

j

t

)

that divides all entries in column

j

t

.

(III)

Finally, adding appropriate

multiples

of row

t

, it can be achieved that all
entries in column

j

t

except for that at position

(

t

,

j

t

)

are zero. This can
be achieved by left-multiplication with an appropriate matrix.

Applying the steps described above to the remaining non-zero columns of the
resulting matrix (if any), we get an

m

×

n

-matrix with column indices

j

1

,

…

,

j

r

where

r

≤

min

⁡

(

m

,

n

)

, each of which satisfies the
following:

1.

the entry at position

(

l

,

j

l

)

is non-zero;

2.

all entries below and above position

(

l

,

j

l

)

as well as entries left of

(

l

,

j

l

)

are zero.

Furthermore, all rows below the

r

-th row are zero.

Now we can re-order the columns of this matrix so that elements on positions

(

i

,

i

)

for

1

≤

i

≤

r

are nonzero and

δ

⁢

(

a

i

,

i

)

≤

δ

⁢

(

a

i

+

1

,

i

+

1

)

for

1

≤

i

<

r

; and all columns right of the

r

-th
column (if present) are zero. For short set

α

i

for the element at
position

(

i

,

i

)

.

δ

has non-negative integer values; so

δ

⁢

(

α

1

)

=

0

is

equivalent

to

α

1

being a unit of

R

.

δ

⁢

(

α

i

)

=

δ

⁢

(

α

i

+

1

)

can either happen if

α

i

and

α

i

+

1

differ by a unit factor, or if they are relatively prime. In the
latter case one can add column

i

+

1

to column

i

(which doesn’t change

α

i

)

and then apply appropriate row manipulations to get

α

i

=

1

.
And for

δ

⁢

(

α

i

)

<

δ

⁢

(

α

i

+

1

)

and

α

i

∤

α

i

+

1

one can apply step (II) after adding column

i

+

1

to column

i

.
This diminishes the minimum

δ

-values for non-zero entries of the matrix,
and by reordering columns etc. we end up with a matrix whose

diagonal

elements

α

i

satisfy

α

i

∣

α

i

+

1

∀

1

≤

i

<

r

.

Since all row and column manipulations

involved in

the process are ,
this shows that there exist

invertible

m

×

m

and

n

×

n

-matrices

S

,

T

so that

S

⁢

A

⁢

T

is

[

α

1

0

…

0

0

α

2

…

0

⋮

⋮

⋱

⋮

0

0

…

α

r

0

0

…

0

⋮

⋮

…

⋮

0

0

…

0

]

.

This is the

Smith normal form

of the matrix. The elements

α

i

are unique up to

associates

and are called

elementary divisors

.

Title

Smith normal form

Canonical name

SmithNormalForm

Date of creation

2013-03-22 13:52:08

Last modified on

2013-03-22 13:52:08

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

21

Author

Mathprof (13753)

Entry type

Topic

Classification

msc 13F10

Defines

elementary divisor

Generated on Fri Feb 9 17:31:30 2018 by

LaTeXML