Matrix Algebras

While a matrix algebra may be defined over any ring R, most non-trivial computations require R to be an Euclidean Domain.

• Arithmetic
• Extension and restriction of coefficient ring
• Direct sum, tensor product
• Determinant (including modular algorithm), trace, characteristic polynomial, minimum polynomial
• Order of a unit (Leedham-Green algorithm)
• Canonical forms over a field: echelon, Jordan, rational, primary rational
• Canonical forms over an ED: echelon, Hermite, Smith
• Characteristic polynomial, minimal polynomial
• Properties of an element: unit, zero-divisor, nilpotent
• Standard basis for subalgebras, left, right and two-sided ideals
• Quotient algebras
• Sum, intersection, product, power of ideal
• Radical of an ideal
• Centre, commutator algebra, Jacobson radical
• Centralizer of a subalgebra in the complete matrix algebra
• Maximal (minimal) left, right, two-sided ideals
• Construction of the (left, right) regular matrix representation

The order of a unit over a finite field is found using the very efficient algorithm of Leedham-Green.