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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,i is in the first quadrant for 0<irankA 

  

Si,i divides Si+1,i+1 for 0<i<rankA 

  

i=1rSi,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+7I&comma;8+10I&comma;68I&comma;5+7I&comma;66I&comma;5I&comma;10+I&comma;13I&comma;10+5I

H−4+7I8+10I−68I−5+7I66I5I−10+I13I−10+5I

(1)

GIsmithH

100010001797+791I

(2)

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

A−48I−110I2+3I−19I8+4I−5+10I

(3)

BGIsmithA&comma;U&comma;V

B100010

(4)

U

−1+4I−1I510I2+3I

(5)

V

043+30I101+8I0−2829I−7521I16621I8999I

(6)

LinearAlgebra:-EqualU·A·V&comma;B

true

(7)

See Also

GaussInt[GIhermite]

LinearAlgebra[HermiteForm]

LinearAlgebra[SmithForm]