Matrix algebras are one of the associative algebra types in Magma. They arise in many areas of which an important one is the representation theory of groups and algebras. Much functionality is provided for computing with individual elements. However, at present some important structural functionality has yet to be implemented.
A general finite-dimensional associative algebra can also be represented either as a matrix algebra or as a finitely-presented algebra. One can move between these representations. Thus, given a finite-dimensional finitely-presented algebra (fp-algebra) one can often construct its regular representation using a non-commutative Gröbner Basis algorithm.
A matrix algebra A is normally specified by giving a set of generating matrices. It can be defined over any exact ring supported by Magma. Given a pair of matrix algebras, functions are provided to construct their direct sum and tensor product algebras.
Constructors are available to create subalgebras and quotient algebras of an ASC-algebra A. However, quotient algebras are only available for the case in which A is defined over a field. If A is an R-algebra and S is a ring such that there exists a monomorphism f : R→S then a function allows the user to construct the S-algebra B obtained by applying f to map the coefficients of elements of A into S. Functions are provided to create the centre of A and the centraliser of a subalgebra of A.
A large number of operations are implemented for the matrix elements. These include properties such as being a unit, nilpotent, or unipotent; invariants such as determinant, trace, order, projective order (when over a finite field); and canonical forms such as Jordan form, rational form, invariant factors (when over a field) and echelon form, Hermite form, and Smith form (when over an Euclidean domain). Other operations include adjoint, characteristic polynomial, minimal polynomial, Hessenberg form, and eigenvalues. The algorithms used for these problems are often very sophisticated so as to reduce the runtime as much as possible.
In order to perform various structural operations, a matrix algebra A over a field K is treated as a vector space M by regarding each n×n matrix as a vector of length n2. Using elementary linear algebra, it is straightforward to construct a basis for A thereby determining its dimension and making it possible to test membership of A. Left-, right- and two-sided ideals can be created. Over a field, a basis is found by a form of spinning algorithm. This allows the user to determine dimension, basis and to test membership of the ideal.
Noting that A has a natural action on M, it can be regarded as an A-module. In the case in which K is a finite field the Meataxe can be applied to M to determine such things as a composition series and the composition factors for A. The minimal left-, right-, or two-sided ideals can also be found. At present no other ideal operations are available.
At present algorithms of Brooksbank and O'Brien are used to compute the Jacobson radical and unit group of A though these will be shortly be replaced by algorithms based on data described in the Section on presentations below. If A is defined over a field that is not a finite field then an algorithm suggested by Willem de Graaf is used.
The functions described in this subsection assume that A is defined over a finite field. A series of algorithms developed by Jon Carlson and Graham Matthews enable the user to construct a presentation for A. In the course of doing this, much important structural information about A is derived:
Once a presentation is available, a non-commutative Gröbner basis algorithm is used to construct a basis for the finitely-presented algebra. In particular, this yields the dimension of the algebra. The presentation machinery also gives a test for membership in a matrix algebra. If A is a subalgebra of the full matrix ring Mat(n,Fq), Magma can decide whether a matrix of Mat(n,Fq) is an element of A. If the matrix is in A then it can be written as a polynomial in the associate algebra defined by the presentation. The construction of a presentation is usually quite fast and algebras of dimension 10,000 or more can be handled.
Let G be the wreath product of the cyclic group of order 2 with the alternating group A7, a group of order 322,560. The matrix algebra A over F2 generated by matrices giving the action of G on the coset space of its Sylow 3-subgroup has degree 1120. Using the Carlson–Matthews algorithms, it takes 68.27 seconds to construct a presentation for A and a further 2.80 seconds to determine that the dimension of A is 77,045. If the same matrix algebra is taken over F5, the runtime is 2363 seconds and the dimension of the algebra is 112,002.