Two Matrix Theorems Arising from Nilpotent Groups

For a nilpotent group $G$ without $\pi$-torsion, and $x$, $y\in G$, if $x^n=y^n$ for a $\pi$-number $n$, then $x=y$; if $x^m{y}^n=y^n{x}^m$ for $\pi$-numbers $m$, $n$, then $xy=yx$. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let $m$ be a given positive integer and $A$ a complex square matrix satisfying that (i) all eigenvalues of $A$ are nonnegative, and (ii) $\mathrm{rank}{A}^2=\mathrm{rank}A$; then $A$ has a unique $m$-th root $X$ with $\mathrm{rank}{X}^2=\mathrm{rank}X$, all eigenvalues of $X$ are nonnegative, and moreover there is a polynomial $f(\lambda )$ with $X=f(A)$. In addition, let $A$ and $B$ be complex $n\times n$ matrices with all eigenvalues nonnegative, and $\mathrm{rank}{A}^2=\mathrm{rank}A$, $\mathrm{rank}{B}^2=\mathrm{rank}B$; then (i) $A=B$ when $A^r=B^r$ for some positive integer $r$, and (ii) $AB=BA$ when $A^s{B}^t=B^t{A}^s$ for two positive integers $s$ and $t$.