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MatrixPolynomialAlgebra

  

SmithForm

  

reduce a Matrix to Smith normal form

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

SmithForm(A, x, out)

Parameters

A

-

Matrix

x

-

(optional) variable

out

-

(optional) equation of the form output = obj where obj is one of 'S', 'U', or 'V', or a list containing one or more of these names; select result objects to compute

Description

• 

The SmithForm(A) command returns the Smith normal form S of a Matrix A with univariate polynomial entries in x over a field F. Thus, the polynomials are regarded as elements of the Euclidean domain F[x].

  

The Smith normal form of a Matrix is a diagonal Matrix S obtained by doing elementary row and column operations. The diagonal entries satisfy the property that for all n <= Rank(A), product(S[i, i], i=1..n) is equal to the (monic) greatest common divisor of all n x n (determinant) minors of A.

• 

The output option (out) determines the content of the returned expression sequence.

  

As determined by the out option, an expression sequence containing one or more of the factors S (the Smith normal form), U (the left-reducing Matrix ), or V (the right-reducing Matrix) is returned. If obj is a list, the objects are returned in the order specified in the list.

  

The returned Matrix objects have the property that S = U . A . V.

• 

Note: The MatrixPolynomialAlgegra:-SmithForm command calls the LinearAlgebra:-SmithForm routine.

Examples

withMatrixPolynomialAlgebra&colon;

AMatrix1&comma;2x&comma;2x2&plus;2x&comma;1&comma;6x&comma;6x2&plus;6x&comma;1&comma;3&comma;x

A:=12x2x2&plus;2x16x6x2&plus;6x13x

(1)

SSmithFormA

S:=10001000x2&plus;32x

(2)

LinearAlgebra&lsqb;Determinant&rsqb;A

8x212x

(3)

lcoeff

x2&plus;32x

(4)

U&comma;VSmithFormA&comma;x&comma;output&equals;&apos;U&apos;&comma;&apos;V&apos;&colon;

mapexpand&comma;U&period;A&period;V

10001000x2&plus;32x

(5)

See Also

expand

indets

lcoeff

LinearAlgebra[Determinant]

LinearAlgebra[Minor]

LinearAlgebra[Rank]

LinearAlgebra[RowOperation]

LinearAlgebra[SmithForm]

map

Matrix

MatrixPolynomialAlgebra

Vector

 


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