# Matrix algebra

algebra of matrices

A subalgebra of the full matrix algebra $F _ {n}$ of all $( n \times n)$- dimensional matrices over a field $F$. The operations in $F _ {n}$ are defined as follows:

$$\lambda a = \| \lambda a _ {ij} \| ,\ \ a + b = \| a _ {ij} + b _ {ij} \| ,$$

$$ab = c = \| c _ {ij} \| ,\ c _ {ij} = \ \sum _ {\nu = 1 } ^ { n } a _ {i \nu } b _ {\nu j }$$

where $\lambda \in F$, and $a = \| a _ {ij} \| , b = \| b _ {ij} \| \in F _ {n}$. The algebra $F _ {n}$ is isomorphic to the algebra of all endomorphisms of an $n$- dimensional vector space over $F$. The dimension of $F _ {n}$ over $F$ equals $n ^ {2}$. Every associative algebra with an identity (cf. Associative rings and algebras) and of dimension over $F$ at most $n$ is isomorphic to some subalgebra of $F _ {n}$. An associative algebra without an identity and with dimension over $F$ less than $n$ can also be isomorphically imbedded in $F _ {n}$. By Wedderburn's theorem, the algebra $F _ {n}$ is simple, i.e. it has only trivial two-sided ideals. The centre of the algebra $F _ {n}$ consists of all scalar $( n \times n)$- dimensional matrices over $F$. The group of invertible elements of $F _ {n}$ is the general linear group $\mathop{\rm GL} ( n, F )$. Every automorphism $h$ of $F _ {n}$ is inner:

$$h( x) = txt ^ {-} 1 ,\ \ x \in F _ {n} ,\ \ t \in \mathop{\rm GL} ( n, F ).$$

Every irreducible matrix algebra (cf. also Irreducible matrix group) is simple. If a matrix algebra $A$ is absolutely reducible (for example, if the field $F$ is algebraically closed), then $A = F _ {n}$ for $n > 1$( 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, $F _ {n}$ contains a unique maximal nilpotent subalgebra — the algebra of all upper-triangular matrices with zero diagonal entries. In $F _ {n}$ there is an $r$- dimensional commutative subalgebra if and only if

$$r \leq \left [ \frac{n ^ {2} }{4} \right ] + 1$$

(Schur's theorem). Over the complex field $\mathbf C$ the set of conjugacy classes of maximal commutative subalgebras of $\mathbf C _ {n}$ is finite for $n < 6$ and infinite for $n > 6$.

In $F _ {n}$ one has the standard identity of degree $2n$:

$$\sum _ {\sigma \in S _ {2 n } } ( \mathop{\rm sgn} \sigma ) x _ {\sigma ( 1) } \dots x _ {\sigma ( 2n) } = 0,$$

where $S _ {2n}$ denotes the symmetric group and $\mathop{\rm sgn} \sigma$ the sign of the permutation $\sigma$, but no identity of lower degree (cf. Amitsur–Levitzki theorem).

How to Cite This Entry:
Matrix algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_algebra&oldid=47793
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article