Irreducible matrix group

A group of -matrices over a field that cannot be brought by simultaneous conjugation in the general linear group to the semi-reduced form

where and are square blocks of fixed dimensions. More accurately, is called irreducible over the field . In the language of transformations: A group of linear transformations of a finite-dimensional vector space is called irreducible if is a minimal -invariant subspace (other than the null space). Irreducible Abelian groups of matrices over an algebraically closed field are one-dimensional. A group of matrices over a field that is irreducible over any extension field is called absolutely irreducible. If is algebraically closed, then for every group the following conditions are equivalent: 1) is irreducible over ; 2) contains matrices that are linearly independent over ; and 3) is absolutely irreducible. Thus, absolute irreducibility over a field is equivalent to irreducibility over the algebraic closure of .

References

[1]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2]	Yu.I. Merzlyakov, "Rational groups" , Moscow (1987) (In Russian)