find the matrix normalizer of a list of matrices

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

Calling Sequences

MatrixNormalizer(M, A)

Parameterss

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

•	The normalizer of a set of matrices contained in a Lie algebra of matrices is the Lie algebra of matrices for all When is a Lie algebra, is an ideal in

•	A list of vectors defining a basis for the normalizer of is returned.

•	For the first calling sequence the normalizer of is calculated in the Lie algebra of all matrices, where is the row dimension of the matrices in

•

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

Example 1.

Find the normalizer of the set of matrices M1.

M1 := [Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0})]

(2.1)

[Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0})]

(2.2)

Example 2.

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

M2 := [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = -1, (1, 2) = -1, (1, 3) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})]