The set of all real, n x n matrices form a Lie algebra with respect to the Lie bracket defined by the matrix commutator [a, b] = ab - ba.  This Lie algebra is usually denoted by gl(n, R).  A matrix Lie algebra is simply a subalgebra of gl(n, R).  Examples of matrix algebras include: [i] the upper triangular n x n matrices; [ii] the strictly upper triangular n x n matrices; [iii] the trace-free n x n matrices; and [iv] the skew-symmetric n x n matrices.  All of these matrix algebras, and many others, can be created with the MatrixAlgebra program.

The Lie algebras of all n x n matrices, the upper triangular n x n matrices, and the strictly upper triangular n x n matrices can be created using the first calling sequence for MatrixAlgebra.  The program returns the required Lie algebra data structure and lists of labels e[i, j] for the vectors and epsilon[i, j] for the dual 1-forms for the matrix Lie algebra to be created.  Here e[i, j] represents the matrix with a 1 in the i-th row and j-th column and zeros elsewhere.

Other matrix algebras are created as subalgebras of gl(n, R), which are symmetries for a list of prescribed tensors using the second calling sequence for MatrixAlgebra.  For example, if T = [t^i_{jk}] is a type (1, 2) tensor on the vector space R^n, then an element a = [a^l_m] of gl(n, R) is a symmetry of T if the equation a^i_m t^m_{jk} - a^l_j t^i_{lk} - a^l_k t^i_{jl} = 0 (sum on l, m) holds.  If we introduce coordinates x^i on R^n, then this symmetry condition is the same as the Lie derivative equation L_X (T) = 0, where T = t^i_{jk} partial_{x^i} dx^j dx^k and X is the linear vector field X = a^l_m x^m partial_{x^l}.  The MatrixAlgebra program, with the keyword option "subalgebra", creates the matrix subalgebra of gl(n, R), which is the symmetry algebra for all the tensors in the list tensorsList.

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