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).

References

[1]	H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) MR0000255 Template:ZBL
[2]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) MR0081264 Template:ZBL
[3]	I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) MR1535024 MR0227205 Template:ZBL
[4]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) MR1541390 Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL
[5]	D.A. Suprunenko, R.I. Tyshkevich, "Commutable matrices" , Minsk (1966) (In Russian)

Comments

A frequently used notation for $ F _ {n} $ is $ M _ {n} ( F ) $.

Wedderburn's theorem on the structure of semi-simple rings says that any semi-simple ring $ R $ is a finite direct product of full matrix rings $ M _ {n _ {i} } ( F _ {i} ) $ over skew-fields $ F _ {i} $, and conversely every ring of this form is semi-simple. Further, the $ F _ {i} $ and $ n _ {i} $ are uniquely determined by $ R $.

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 MR0530404 Template:ZBL