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:
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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:
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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
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(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
:
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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 |
Matrix algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_algebra&oldid=13522