find the matrix centralizer of a list of matrices - Maple Programming Help

Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : DifferentialGeometry/LieAlgebras/MatrixCentralizer

LieAlgebras[MatrixCentralizer] - find the matrix centralizer of a list of matrices

Calling Sequences

MatrixCentralizer(M)

Parameters

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

Description

 • 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(...).

Examples

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

Example 1.

Find the Matrix centralizer of the set of matrices M1.

 > $\mathrm{M1}≔\left[\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right)\right]$
 ${\mathrm{M1}}{≔}\left[\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]\right]$ (2.1)
 > $\mathrm{MatrixCentralizer}\left(\mathrm{M1}\right)$
 $\left[\left[\begin{array}{rr}{1}& {0}\\ {0}& {1}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]\right]$ (2.2)

Example 2.

Find the Matrix centralizer of the set of matrices M2.

 > $\mathrm{M2}≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[0,0,1\right],\left[0,0,0\right],\left[0,0,0\right]\right],\left[\left[0,0,1\right],\left[0,0,1\right],\left[0,0,0\right]\right],\left[\left[-1,-1,0\right],\left[0,-1,0\right],\left[0,0,0\right]\right]\right]\right)$
 ${\mathrm{M2}}{≔}\left[\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{-}{1}& {-}{1}& {0}\\ {0}& {-}{1}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]$ (2.3)
 > $\mathrm{MatrixCentralizer}\left(\mathrm{M2}\right)$
 $\left[\left[\begin{array}{rrr}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {1}\end{array}\right]\right]$ (2.4)