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RegularChains[MatrixTools]

  

LowerEchelonForm

  

lower echelon form of a matrix modulo a regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

LowerEchelonForm(A, rc, R)

Parameters

A

-

square Matrix with coefficients in the ring of fractions of R

rc

-

regular chain of R

R

-

polynomial ring

Description

• 

The command LowerEchelonForm(A, rc, R) returns a list of pairs Bi,rci where rci is a regular chain, and Bi is the lower echelon form of A modulo the saturated ideal of rc_i.

• 

All the returned regular chains rci form a triangular decomposition of rc (in the sense of Kalkbrener).

• 

It is assumed that rc is strongly normalized.

• 

The algorithm is an adaptation of the algorithm of Bareiss.

• 

This command is part of the RegularChains[MatrixTools] package, so it can be used in the form LowerEchelonForm(..) only after executing the command with(RegularChains[MatrixTools]).  However, it can always be accessed through the long form of the command by using RegularChains[MatrixTools][LowerEchelonForm](..).

Examples

withRegularChains:withChainTools:withMatrixTools:

RPolynomialRingx,y,z

R:=polynomial_ring

(1)

TEmptyR:

TChainz+1z+2,y2+z,xzxy,T,R

T:=regular_chain

(2)

EquationsT,R

x2+yzx+zy,y2+z,z2+3z+2

(3)

mMatrixx,y,z,x+1,y+2,z+3,x+4,y+5,z+6

m:=xyzx+1y+2z+3x+4y+5z+6

(4)

lemLowerEchelonFormm,T,R

lem:=600030x+4y+5z+6,regular_chain,1200630x+4y+5z+6,regular_chain,000630x+4y+5z+6,regular_chain,3x+6y+6003x3y+30x+4y+5z+6,regular_chain

(5)

See Also

Chain

Empty

Equations

IsStronglyNormalized

IsZeroMatrix

JacobianMatrix

MatrixInverse

MatrixMultiply

MatrixOverChain

MatrixTools

NormalForm

PolynomialRing

RegularChains

 


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