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RegularChains[MatrixTools]

  

MatrixInverse

  

compute the inverse of a matrix modulo a regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

MatrixInverse(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 MatrixInverse(A, rc, R) returns two lists.

• 

The first list the command returns is a list of pairs Bi,rci where rci is a regular chain and Bi is the inverse of A modulo the saturated ideal of rci.

• 

The second list the command returns is a list of triplets noInv,A,rci where rci is a regular chain and A is the input matrix such that A is not invertible modulo the saturated ideal of rci.

• 

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 MatrixInverse(..) 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][MatrixInverse](..).

Examples

Automatic case discussion.

withRegularChains:withChainTools:withMatrixTools:

RPolynomialRingy,z;rcEmptyR

Rpolynomial_ring

rcregular_chain

(1)

Assume we have two variables y and z that have the same square and z is a 4th root of -1. Suppose we need to compute modulo this relation.

rcChainz4+1,y2z2,rc,R:Equationsrc,R

y2z2,z4+1

(2)

mMatrix1,y+z,0,yz

m1y+z0yz

(3)

We want to compute the inverse of the previous matrix.

mimMatrixInversem,rc,R

mim10012z3,regular_chain,noInv,1y+z0yz,regular_chain

(4)

Let us check the first result.

m1mim111;rc1mim112;Equationsrc1,R

m110012z3

rc1regular_chain

y+z,z4+1

(5)

MatrixMultiplym1,m,rc1,R

1001

(6)

Consider now this other matrix.

mMatrix1,y+z,2,yz

m1y+z2yz

(7)

mimMatrixInversem,rc,R

mim10z312z3,regular_chain,01212z314z3,regular_chain,

(8)

m1mim111;rc1mim112

m110z312z3

rc1regular_chain

(9)

m2mim121;rc2mim122

m201212z314z3

rc2regular_chain

(10)

MatrixMultiplym2,m,rc2,R

1001

(11)

MatrixMultiplym2,m,rc2,R

1001

(12)

Get a generic answer that would hold both cases.

clrMatrixCombinerc1,rc2,R,m1,m2

clr12yz3+1214yz3+1414z2y34z318z2y+38z3,regular_chain

(13)

Check.

MatrixMultiplyclr11,m,clr12,R

1001

(14)

See Also

Chain

Empty

Equations

IsStronglyNormalized

IsZeroMatrix

JacobianMatrix

LowerEchelonForm

MatrixCombine

MatrixMultiply

MatrixOverChain

MatrixTools

NormalForm

PolynomialRing

RegularChains