MatrixNormalizer - Maple Help
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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

 • The normalizer of a set of matrices contained in a Lie algebra of matrices is the Lie algebra of matrices for all When $M$ is a Lie algebra, ${\mathbf{nor}}_{A}\left(M\right)$is an ideal in
 • A list of matrices defining a basis for the normalizer of is returned.
 • For the first calling sequence the normalizer of M is calculated in the Lie algebra of all matrices, where 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

Find the normalizer of the set of matrices M1.

 > $\mathrm{M1}≔\left[\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right)\right]$
 > $\mathrm{MatrixNormalizer}\left(\mathrm{M1}\right)$

Example 2.

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

 > $\mathrm{M2}≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[0,0,1\right],\left[0,0,0\right],\left[0,0,0\right]\right],\left[\left[0,0,1\right],\left[0,0,1\right],\left[0,0,0\right]\right],\left[\left[-1,-1,0\right],\left[0,-1,0\right],\left[0,0,0\right]\right]\right]\right)$
 > $A≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[1,0,0\right],\left[0,0,0\right],\left[0,0,0\right]\right],\left[\left[0,1,0\right],\left[0,0,0\right],\left[0,0,0\right]\right],\left[\left[0,0,1\right],\left[0,0,0\right],\left[0,0,0\right]\right],\left[\left[0,0,0\right],\left[0,1,0\right],\left[0,0,0\right]\right],\left[\left[0,0,0\right],\left[0,0,1\right],\left[0,0,0\right]\right],\left[\left[0,0,0\right],\left[0,0,0\right],\left[0,0,1\right]\right]\right]\right)$
 > $N≔\mathrm{MatrixNormalizer}\left(\mathrm{M2},A\right)$

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

 > $\mathrm{LieAlgebraData}\left(N\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}\right]$ (2.1)