Irreducible matrix group

A group $G$ of $n \times n$-matrices over a field $k$ that cannot be brought by simultaneous conjugation in the general linear group $\mathrm{GL}(n,k)$ to the semi-reduced form $$ \left( \begin{array}{cc} A & \star \\ 0 & B \end{array} \right) $$ where $A$ and $B$ are square blocks of fixed dimensions. More accurately, $G$ is called irreducible over the field $k$. In the language of transformations: A group $G$ of linear transformations of a finite-dimensional vector space $V$ is called irreducible if $V$ is a minimal $G$-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 $k$ is algebraically closed, then for every group $G \subseteq \mathrm{GL}(n,k)$ the following conditions are equivalent: 1) $G$ is irreducible over $k$; 2) $G$ contains $n^2$ matrices that are linearly independent over $k$; and 3) $G$ is absolutely irreducible. Thus, absolute irreducibility over a field $k$ is equivalent to irreducibility over the algebraic closure of $k$.

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)