MINIMALLY IRREDUCIBLE FAMILIES OF COMPLEX MATRICES

Abstract A family of n times n $n\times n$ n × n matrices over a field double struck upper F $\mathbb {F}$ 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 double struck upper F $\mathbb {F}$ F is algebraically closed and n greater than or equals 2 $n\ge 2$ n ≥ 2 , a minimally irreducible family has at most 2 n minus 1 $2n-1$ 2 n − 1 elements. We show that for complex n times n $n\times n$ n × n matrices, n greater than or equals 3 $n\ge 3$ n ≥ 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 less than or equals k less than or equals 2 n minus 2 $n\le k\le 2n-2$ n ≤