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Hermite

compute the Hermite Normal Form of a matrix mod p

Smith

compute the Smith Normal Form of a matrix mod p

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Hermite(A, x)

Hermite(A, x, U)

Smith(A, x, U, V)

Parameters

A

-

rectangular matrix of univariate polynomials in x

x

-

name

U

-

name

V

-

name

Description

• 

Hermite and Smith are placeholders for representing the Hermite and Smith Normal Forms respectively. They are used in conjunction with mod as is described below.

• 

Hermite(A, x) mod p computes the Hermite Normal Form (reduced row echelon form) of an m by n rectangular matrix of univariate polynomials in x over the integers modulo p. The polynomial coefficients must be rational or elements of a finite extension field specified by RootOfs. In the case of three arguments, the third argument, U, will be assigned the transformation matrix upon completion, such that Hermite(A) = U &* A.

• 

Smith(A, x) mod p computes the Smith Normal Form of a matrix with univariate polynomial entries in x over the integers modulo p. The coefficients of the polynomial must be either rational or elements of a finite extension field specified by RootOfs. 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.

Examples

AMatrix1+x,1+x2,1+x2,1+x4

A:=1+xx2+1x2+1x4+1

(1)

SmithA,x,U,Vmod2

1+x00x4+x3+x2+x

(2)

evalU

10x3+11

(3)

evalV

x1+x11

(4)

restart

AMatrix1+x,1+x2,1+x2,1+x4:HermiteA,x,Umod2

1+xx2+10x4+x3+x2+x

(5)

evalU

101+x1

(6)

To obtain the column form for the HNF of A do

LinearAlgebra[Transpose]HermiteLinearAlgebra[Transpose]A,xmod2

1+x0x2+1x4+x3+x2+x

(7)

See Also

LinearAlgebra[HermiteForm]

LinearAlgebra[Modular]

LinearAlgebra[SmithForm]

RootOf

 


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