On m-th roots of complex matrices

Abstract

For an $n\times n$ matrix $M$, $\sigma(M)$ denotes the set of all different eigenvalues of $M$. In this paper, we will prove two results on the $m$-th $(m\geq2)$ roots of a matrix $A$. Firstly, let $X$ be an $m$-th root of $A$. Then $X$ can be expressed as a polynomial in $A$ if and only if rank $X^2$= rank $X$ and $|\sigma(X)|=|\sigma(A)|$. Secondly, let $X$ and $Y$ be two $m$-th roots of $A$. If both $X$ and $Y$ can be expressed as polynomials in $A$, then $X=Y$ if and only if $\sigma(X)=\sigma(Y)$.