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LieAlgebras[MatrixNormalizer] - find the matrix normalizer of a list of matrices

Calling Sequences

     MatrixNormalizer(M, A)

Parameters

     M   - a list of square matrices, each of the same dimension

     A   - (optional) a list of square matrices, each of the same dimension, containing the matrices M, and forming a Lie algebra

 

 

Description

Examples

Description

• 

The normalizer of a set of matrices M contained in a Lie algebra of matrices A is the Lie algebra of matrices norAM = {a  A |  ab  ba M for all b M}. When M is a Lie algebra, norAMis an ideal in A. 

• 

A list of matrices defining a basis for the normalizer of M is returned.

• 

For the first calling sequence the normalizer of M is calculated in the Lie algebra of all n ×n matrices, where n is the row dimension of the matrices in M.

• 

The command MatrixNormalizer is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form MatrixNormalizer(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MatrixNormalizer(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Find the normalizer of the set of matrices M1.

M1Matrix0,1,0,0

M1:=0100

(2.1)

MatrixNormalizerM1

0001,0100,1000

(2.2)

 

Example 2.

Find the normalizer of the set of matrices M2 within the Lie algebra A.

M2mapMatrix,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,0

M2:=001000000,001001000,110010000

(2.3)

AmapMatrix,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1

A:=100000000,010000000,001000000,000010000,000001000,000000001

(2.4)

NMatrixNormalizerM2,A

N:=000000001,000001000,100010000,001000000,010000000

(2.5)

 

We use the LieAlgebraData command to calculate the commutation relations for the Lie algebra of matrices N.

LieAlgebraDataN

e1,e2=e2,e1,e4=e4,e2,e3=e2,e2,e5=e4,e3,e4=e4

(2.6)

See Also

DifferentialGeometry

LieAlgebras

SubalgebraNormalizer