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GaussInt

  

GIsmith

  

Gaussian Integer-only Smith Normal Form

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

GIsmith(A)

GIsmith(A, U, V)

Parameters

A

-

Matrix of Gaussian integers

U

-

name (optional)

V

-

name (optional)

Description

• 

The function GIsmith computes the Smith normal form S of an n by m Matrix of Gaussian integers.

• 

If two n by n Matrices have the same Smith normal form, they are equivalent.

• 

The Smith normal form is a diagonal Matrix S where

  

rankA = number of nonzero rows (columns) of S

  

Si&comma;i is in the first quadrant for 0<irankA 

  

Si&comma;i divides Si&plus;1&comma;i&plus;1 for 0<irankA 

  

i&equals;1rSi&comma;i divides detM for all minors M of rank  0<rrankA 

• 

The Smith normal form is obtained by doing elementary row and column operations.  This includes interchanging rows (columns), multiplying through a row (column) by a unit in Zi, and adding integral multiples of one row (column) to another.

• 

In the case of three arguments, the second argument U and the third argument V will be assigned the transformation Matrices on output, such that GIsmith(A) = U . A . V.

Examples

withGaussInt&colon;

HMatrix4&plus;7I&comma;8&plus;10I&comma;68I&comma;5&plus;7I&comma;66I&comma;5I&comma;10&plus;I&comma;13I&comma;10&plus;5I

H:=4&plus;7I8&plus;10I68I5&plus;7I66I5I10&plus;I13I10&plus;5I

(1)

GIsmithH

100010001797&plus;791I

(2)

AMatrix48I&comma;110I&comma;2&plus;3I&comma;19I&comma;8&plus;4I&comma;5&plus;10I

A:=48I110I2&plus;3I19I8&plus;4I5&plus;10I

(3)

BGIsmithA&comma;U&comma;V

B:=100010

(4)

U

1&plus;4I1I510I2&plus;3I

(5)

V

043&plus;30I101&plus;8I02829I7521I16621I8999I

(6)

LinearAlgebra:-EqualU&period;A&period;V&comma;B

true

(7)

See Also

GaussInt[GIhermite]

LinearAlgebra[HermiteForm]

LinearAlgebra[SmithForm]

 


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