The normalizer of a set of matrices $M$contained in a Lie algebra of matrices $A$is the Lie algebra of matrices ${\mathbf{nor}}_A\left(M\right) \=$$\{a \in A \| a\cdot b - b\cdot a \in M$for all $b \in M\}\.$When $M$ is a Lie algebra, ${\mathbf{nor}}_A\left(M\right)$is 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 $n \times n$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(...).

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

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

$\mathrm{MatrixNormalizer}\left(\mathrm{M1}\right)$

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

$A≔\mathrm{map}\left(\mathrm{Matrix}\,\left[\left[\left[1\,0\,0\right]\,\left[0\,0\,0\right]\,\left[0\,0\,0\right]\right]\,\left[\left[0\,1\,0\right]\,\left[0\,0\,0\right]\,\left[0\,0\,0\right]\right]\,\left[\left[0\,0\,1\right]\,\left[0\,0\,0\right]\,\left[0\,0\,0\right]\right]\,\left[\left[0\,0\,0\right]\,\left[0\,1\,0\right]\,\left[0\,0\,0\right]\right]\,\left[\left[0\,0\,0\right]\,\left[0\,0\,1\right]\,\left[0\,0\,0\right]\right]\,\left[\left[0\,0\,0\right]\,\left[0\,0\,0\right]\,\left[0\,0\,1\right]\right]\right]\right)$

$N≔\mathrm{MatrixNormalizer}\left(\mathrm{M2}\,A\right)$

$\mathrm{LieAlgebraData}\left(N\right)$

$\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=-\mathrm{e4}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=-\mathrm{e4}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e4}\right]$