MINIMALLY IRREDUCIBLE FAMILIES OF COMPLEX MATRICES

Abstract A family of $n\times n$ matrices over a field $\mathbb {F}$ is irreducible if it has no common nontrivial invariant subspace, and minimally irreducible if it is irreducible but has no proper irreducible subfamily. If $\mathbb {F}$ is algebraically closed and $n\ge 2$ , a minimally irreducible family has at most $2n-1$ elements. We show that for complex $n\times n$ matrices, $n\ge 3$ , a family of minimally irreducible (i) matrix units, (ii) rank one projections, (iii) unicellular matrices and (iv) orthoatomic matrices has k elements where respectively (i) $n\le k\le 2n-2$ , (ii) $k=n$ , (iii) $2\le k\le n-1$ and (iv) $2\le k\le n-1$ . All of the values of k in these ranges are attained. If $n=2$ , each such minimally irreducible family has $2$ elements.