Matrix algebra

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.

References

[1]	H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946)
[2]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[3]	I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)
[4]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[5]	D.A. Suprunenko, R.I. Tyshkevich, "Commutable matrices" , Minsk (1966) (In Russian)

Comments

A frequently used notation for is .

Wedderburn's theorem on the structure of semi-simple rings says that any semi-simple ring is a finite direct product of full matrix rings over skew-fields , and conversely every ring of this form is semi-simple. Further, the and are uniquely determined by .

The Wedderburn–Artin theorem says that a right Artinian simple ring is a total matrix ring (E. Artin, 1928; proved for finite-dimensional algebras by J.H.M. Wedderburn in 1907). A far-reaching generalization of this is the Jacobson density theorem, cf. Associative rings and algebras and [a1].

References

[a1]	P.M. Cohn, "Algebra" , 2 , Wiley (1977) pp. Sect. 10.2