View source for Matrix algebra

''algebra of matrices''

A subalgebra of the full matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627901.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627902.png" />-dimensional matrices over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627903.png" />. The operations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627904.png" /> are defined as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627905.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627906.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627907.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627908.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m0627909.png" /> is isomorphic to the algebra of all endomorphisms of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279010.png" />-dimensional vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279011.png" />. The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279012.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279013.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279014.png" />. Every associative algebra with an identity (cf. [[Associative rings and algebras|Associative rings and algebras]]) and of dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279015.png" /> at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279016.png" /> is isomorphic to some subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279017.png" />. An associative algebra without an identity and with dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279018.png" /> less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279019.png" /> can also be isomorphically imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279020.png" />. By Wedderburn's theorem, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279021.png" /> is simple, i.e. it has only trivial two-sided ideals. The centre of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279022.png" /> consists of all scalar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279023.png" />-dimensional matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279024.png" />. The group of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279025.png" /> is the [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279026.png" />. Every [[Automorphism|automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279028.png" /> is inner:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279029.png" /></td> </tr></table>

Every irreducible matrix algebra (cf. also [[Irreducible matrix group|Irreducible matrix group]]) is simple. If a matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279030.png" /> is absolutely reducible (for example, if the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279031.png" /> is algebraically closed), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279033.png" /> (Burnside's theorem). A matrix algebra is semi-simple if and only if it is completely reducible (cf. also [[Completely-reducible matrix group|Completely-reducible matrix group]]).

Up to conjugation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279034.png" /> contains a unique maximal nilpotent subalgebra — the algebra of all upper-triangular matrices with zero diagonal entries. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279035.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279036.png" />-dimensional commutative subalgebra if and only if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279037.png" /></td> </tr></table>

(Schur's theorem). Over the complex field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279038.png" /> the set of conjugacy classes of maximal commutative subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279039.png" /> is finite for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279040.png" /> and infinite for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279041.png" />.

In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279042.png" /> one has the standard identity of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279043.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279044.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279045.png" /> denotes the [[Symmetric group|symmetric group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279046.png" /> the sign of the permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279047.png" />, but no identity of lower degree (cf. [[Amitsur–Levitzki theorem]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) {{MR|0081264}} {{ZBL|0073.02002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) {{MR|1535024}} {{MR|0227205}} {{ZBL|0177.05801}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German) {{MR|1541390}} {{ZBL|1032.00002}} {{ZBL|1032.00001}} {{ZBL|0903.01009}} {{ZBL|0781.12003}} {{ZBL|0781.12002}} {{ZBL|0724.12002}} {{ZBL|0724.12001}} {{ZBL|0569.01001}} {{ZBL|0534.01001}} {{ZBL|0997.00502}} {{ZBL|0997.00501}} {{ZBL|0316.22001}} {{ZBL|0297.01014}} {{ZBL|0221.12001}} {{ZBL|0192.33002}} {{ZBL|0137.25403}} {{ZBL|0136.24505}} {{ZBL|0087.25903}} {{ZBL|0192.33001}} {{ZBL|0067.00502}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D.A. Suprunenko, R.I. Tyshkevich, "Commutable matrices" , Minsk (1966) (In Russian)</TD></TR></table>



====Comments====
A frequently used notation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279048.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279049.png" />.

Wedderburn's theorem on the structure of semi-simple rings says that any semi-simple ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279050.png" /> is a finite direct product of full matrix rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279051.png" /> over skew-fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279052.png" />, and conversely every ring of this form is semi-simple. Further, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279054.png" /> are uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062790/m06279055.png" />.

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|Associative rings and algebras]] and [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''2''' , Wiley (1977) pp. Sect. 10.2 {{MR|0530404}} {{ZBL|0341.00002}} </TD></TR></table>