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hermite

  

Hermite Normal Form (reduced row echelon form)

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

hermite(A, x)

hermite(A, x, U)

Parameters

A

-

rectangular matrix of polynomials in x

x

-

name

U

-

name

Description

• 

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

  

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

• 

The function hermite computes the Hermite Normal Form (reduced row echelon form) of an m by n rectangular matrix of univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q.

• 

In principle this should work for polynomials in x over any field F, i.e. the Euclidean domain F[x], but in practice the code is only as powerful as Maple's normal function.

• 

The Hermite normal form is obtained by doing elementary row operations on A.  This includes interchanging rows, multiplying through a row by a unit, and subtracting a multiple of one row from another.

• 

One can use transposes to obtain the column form of the Hermite Normal Form of a matrix.

• 

In the case of three arguments, the third argument U will be assigned the transformation matrix on output, such that the following holds: hermite(A) = U &* A.

• 

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

Examples

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

withlinalg:

Hinversehilbert2,x

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

(1)

hermiteH,x

x25x+600x27x+12

(2)

Hmatrix3,3,1,x,x2,x3+2x1,0,1,1,x1,x2+1

H1xx2x3+2x1011x1x2+1

(3)

hermiteH,x

10x2+x01−100x5+x4+2x3+x2x1

(4)

To obtain the column form of HNF for H do

transposehermitetransposeH,x

100010x4x32x2x+2x+1x5+x4+2x3+x2x1

(5)

Amatrix2,3,x+1,x2+1,2x1,x+2,x2+x,x2:

BhermiteA,x,U

B1012x212x2x1+x22+122+x01x2+12x1+x2122+x

(6)

evalU

12x212xx22+12x2+1x212

(7)

mapnormal,evalmU &* AB

000000

(8)

See Also

Hermite

linalg(deprecated)[ihermite]

linalg(deprecated)[smith]

LinearAlgebra

LinearAlgebra[HermiteForm]

Smith