The centralizer of a set of matrices M is the Lie algebra of matrices which commute with all the matrices in M.

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

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

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

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

$\mathrm{M1}≔\left[\left[\begin{array}{rr}0& 1\\ 0& 0\end{array}\right]\right]$

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

$\left[\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\,\left[\begin{array}{rr}0& 1\\ 0& 0\end{array}\right]\right]$

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

$\mathrm{M2}≔\left[\left[\begin{array}{rrr}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\,\left[\begin{array}{rrr}0& 0& 1\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]\,\left[\begin{array}{rrr}-1& -1& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right]\right]$

$\mathrm{MatrixCentralizer}\left(\mathrm{M2}\right)$

$\left[\left[\begin{array}{rrr}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\right]$