Abstract A family of n n matrices over a field 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 F is algebraically closed and n 2, a minimally irreducible family has at most 2n-1 elements. We show that for complex n n matrices, 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 k 2n-2, (ii) k=n, (iii) 2 k n-1 and (iv) 2 k n-1. All of the values of k in these ranges are attained. If n=2, each such minimally irreducible family has 2 elements.
W. E. LONGSTAFF (Fri,) studied this question.