Let $\displaystyle F$ be a field and let $\displaystyle F_n$ be the ring of all $\displaystyle n\times n$ matrices over $\displaystyle F.$ We consider $\displaystyle F_n$ as the ring of all linear transformations on the vector space $\displaystyle V$ of $\displaystyle n-$tuples of elements of $\displaystyle F.$ If $\displaystyle A$ is a subset of $\displaystyle F_n$ let $\displaystyle \overline{A}$ be the subalgebra generated by $\displaystyle A$ over $\displaystyle F.$ Clearly $\displaystyle V$ is a faithful $\displaystyle F_n,$ and so, $\displaystyle \overline{A}-$module. $\displaystyle V$ is, in addition, both a unitary and irreducible $\displaystyle F_n-$module.

We say the set of matrices $\displaystyle A$ is irreducible if $\displaystyle V$ is an irreducible $\displaystyle \overline{A}-$module. In matrix terms this merely says that there is no invertible matrix $\displaystyle S$ in $\displaystyle F_n$ so that

$\displaystyle S^{-1}aS=\left(\begin{array}{c|c} a_1 & 0 \\ \hline * & a_2 \end{array}\right) $

for all $\displaystyle a\in A.$