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linalg(deprecated)

  

smith

  

compute the Smith normal form of a matrix

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

smith(A, x)

smith(A, x, U, V)

Parameters

A

-

square matrix of univariate polynomials in x

x

-

the variable name

U

-

name

V

-

name

Description

• 

Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[SmithForm], instead.

  

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

• 

The Smith normal form of a matrix with univariate polynomial entries in x over a field F is computed. Thus the polynomials are then regarded as elements of the Euclidean domain F[x].

• 

This routine is only as powerful as Maple's normal function, since at present it only understands the field Q of rational numbers and rational functions over Q.

• 

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 following property for all nrankA: i=1nSi,i is equal to the (monic) greatest common divisor of all n by n minors of A.

• 

In the case of four arguments, the third argument U and the fourth argument V will be assigned the transformation matrices on output, such that smith(A) = U &* A &* V.

• 

The command with(linalg,smith) allows the use of the abbreviated form of this command.

Examples

Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[SmithForm], instead.

withlinalg:

A1xyxy01x2

A1xxy+y0x2+1

(1)

smithA,x

1+x00x21

(2)

Hinversehilbert2,x

H3+x22+x3+x2+x4+x3+x2+x4+x3+x24+x

(3)

smithH,x

3+x00x39x2+26x24

(4)

BsmithA,x,U,V

B1+x00x21

(5)

evalU

−10x+1−1

(6)

evalV

1yy11

(7)

evalmU &* A &* VB

1+x1y+xyy+1x1+xy+xyyx+11x1y+x+1xy+y+x21x+11xy+x+1xy+y

(8)

See Also

linalg(deprecated)[hermite]

linalg(deprecated)[ismith]

LinearAlgebra

LinearAlgebra[SmithForm]

Smith