We show that an irreducible family ${\mathcal{S}}$ of complex $n\times n$ matrices satisfies Paz’s conjecture if it contains a rank-one matrix. We next investigate properties of families of rank-one matrices. If ${\mathcal{R}}$ is a linearly independent, irreducible family of rank-one matrices then (i) ${\mathcal{R}}$ has length at most $n$, (ii) if all pairwise products are nonzero, ${\mathcal{R}}$ has length 1 or 2, (iii) if ${\mathcal{R}}$ consists of elementary matrices, its minimum spanning length $M$ is the smallest integer $M$ such that every elementary matrix belongs to the set of words in ${\mathcal{R}}$ of length at most $M$. Finally, for any integer $k$ dividing $n-1$, there is an irreducible family of elementary matrices with length $k+1$.

中文翻译：

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包含秩一矩阵的复矩阵的不可约族

我们证明了一个不可约的家庭${\mathcal{S}}$复杂的$n\次 n$如果矩阵包含秩一矩阵，则矩阵满足 Paz 猜想。我们接下来研究一级矩阵族的性质。如果${\mathcal{R}}$是一个线性独立的、不可约的一阶矩阵族，则 (i)${\mathcal{R}}$最多有长度$n$, (ii) 如果所有成对乘积均非零，${\mathcal{R}}$长度为 1 或 2，(iii) 如果${\mathcal{R}}$由基本矩阵组成，其最小跨越长度$M$是最小的整数$M$使得每个基本矩阵都属于${\mathcal{R}}$最多长度$M$. 最后，对于任何整数$k$划分$n-1$，有一个不可约的基本矩阵族，其长度为$k+1$.