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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
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Description
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A list of vectors defining a basis for the normalizer of  is returned.
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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(...).
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Examples
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Example 1.
Find the normalizer of the set of matrices M1.
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![M1 := [Matrix([[0, 1], [0, 0]])]](/support/helpjp/helpview.aspx?si=6623/file05816/math83.png)
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![M1 := [Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0})]](/support/helpjp/helpview.aspx?si=6623/file05816/math86.png)
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![[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})]](/support/helpjp/helpview.aspx?si=6623/file05816/math93.png)
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Example 2.
Find the normalizer of the set of matrices M2 within the Lie algebra A.
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![M2 := map(Matrix, [[[0, 0, 1], [0, 0, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 1], [0, 0, 0]], [[-1, -1, 0], [0, -1, 0], [0, 0, 0]]])](/support/helpjp/helpview.aspx?si=6623/file05816/math106.png)
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![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})]](/support/helpjp/helpview.aspx?si=6623/file05816/math109.png)
| (2.3) |
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![A := map(Matrix, [[[1, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 1, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 1, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 1]]])](/support/helpjp/helpview.aspx?si=6623/file05816/math113.png)
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| (2.4) |
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| (2.5) |
We use the LieAlgebraData command to calculate the commutation relations for the Lie algebra of matrices 
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![_DG([["LieAlgebra", "L1", [5]], [[[1, 2, 2], -1], [[1, 4, 4], -1], [[2, 3, 2], -1], [[2, 5, 4], -1], [[3, 4, 4], 1]]])](/support/helpjp/helpview.aspx?si=6623/file05816/math144.png)
| (2.6) |
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