# Matrix algebra

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algebra of matrices

A subalgebra of the full matrix algebra of all -dimensional matrices over a field . The operations in are defined as follows:  where , and . The algebra is isomorphic to the algebra of all endomorphisms of an -dimensional vector space over . The dimension of over equals . Every associative algebra with an identity (cf. Associative rings and algebras) and of dimension over at most is isomorphic to some subalgebra of . An associative algebra without an identity and with dimension over less than can also be isomorphically imbedded in . By Wedderburn's theorem, the algebra is simple, i.e. it has only trivial two-sided ideals. The centre of the algebra consists of all scalar -dimensional matrices over . The group of invertible elements of is the general linear group . Every automorphism of is inner: Every irreducible matrix algebra (cf. also Irreducible matrix group) is simple. If a matrix algebra is absolutely reducible (for example, if the field is algebraically closed), then for (Burnside's theorem). A matrix algebra is semi-simple if and only if it is completely reducible (cf. also Completely-reducible matrix group).

Up to conjugation, contains a unique maximal nilpotent subalgebra — the algebra of all upper-triangular matrices with zero diagonal entries. In there is an -dimensional commutative subalgebra if and only if (Schur's theorem). Over the complex field the set of conjugacy classes of maximal commutative subalgebras of is finite for and infinite for .

In one has the standard identity of degree : where denotes the symmetric group and the sign of the permutation , but no identity of lower degree.

How to Cite This Entry:
Matrix algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_algebra&oldid=13522
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article