$m$-幂零-清洁非奇异矩阵
A bound for the nilpotence index associated to $m$-nil-clean nonderogatory matrices
AI总结：
研究在特定条件下，$n\times n$非奇异矩阵$A$的分解问题，通过证明得出存在$m$个幂等矩阵和一个幂零矩阵使$A$可分解，且给出幂零矩阵幂零指数及$n > p$时$A$按$n$奇偶性的不同分解形式。
AI中文摘要：
证明了若$\mathbb{F}$是特征为$p$的正域，$m$和$n$是正整数且$m\geq2$，$n\leq p\leq mn - 1$，对于每个迹在$\{k\cdot1_{\mathbb{F}}\mid k\in\{0,1,\dots,p - 1\}\}$的$n\times n$非奇异矩阵$A\in\mathbb{M}_n(\mathbb{F})$，存在$m$个幂等矩阵$E_1,E_2,\dots,E_m$和一个幂零矩阵$N$，使得$A = E_1 + E_2+\dots+E_m + N$，并给出了$N$的幂零指数$k$的具体情况。此外，对于$n > p$，根据$n$的奇偶性给出了$A$的分解形式。
英文摘要：
It is proved that if $\mathbb{F}$ is a field of positive characteristic $p,$ and if $m$ and $n$ are positive integers such that $m\geq2,$ and $4\leq n\leq p\leq mn-1,$ for every $n\times n$ nonderogatory matrix $A\in \mathbb{M}_n(\mathbb{F}),$ with trace in $\{k.1_{\mathbb{F}}\mid k\in \{0,1,\dots,p-1\}\},$ there exist $m$ idempotent matrices $E_1, E_2,\dots, E_m,$ and a nilpotent matrix $N$, such that $A=E_1+E_2+\dots+E_m+N,$ with $N^k=0,$ where $k=n$ if $p\in \{nm-1,nm-2\},$ $k=n-1$ if $p=nm-3,$ otherwise $k=\mathrm{max}(2,1+\lfloor\frac{n-1}{r}\rfloor),$ if $n$ is even, and $k=\mathrm{max}(3,1+\lfloor\frac{n-1}{r}\rfloor),$ if $n$ is odd, where $r:=\lfloor\frac{nm-p}{2}\rfloor.$ Moreover, for $4\leq n>p,$ $A$ is the sum of two idempotent matrices, and a square zero one, if $n$ is even, and it is sum of two idempotent matrices and one whose third power is zero, if $n$ is odd.
