Let $m\in\mathbb{N}$ and set $\mathbf{B}:=\mathbf{A}^{m}$, where $\mathbf{A}$ is a nonsingular matrix (formally $\mathbf{A}=\sqrt[m]{\mathbf{B}}$). If we know the eigenvalues of $\mathbf{B}$, what can we say about the eigenvalues of $\mathbf{A}$? It is obvious that if $\lambda$ is an eigenvalue of $\mathbf{A}$, then $\lambda^{m}$ is an eigenvalue of $\mathbf{B}$. But I need the reverse implication. For instance, do we have the following relations?
- $\sigma(\mathbf{A})\subset\sqrt[m]{\sigma(\mathbf{B})}=\{\sqrt[m]{\omega}:\ \omega\in\sigma(\mathbf{B})\}$ (spectrum).
- $\rho(\mathbf{A})=\sqrt[m]{\rho(\mathbf{B})}=\sqrt[m]{\max\{|\omega|:\ \omega\in\sigma(\mathbf{B})\}}$ (spectral radius).
I would be very glad if you can give a reference for relations concerning the properties above.
Suppose $c$ is an eigenvalue of $B$, $A^m(x)=B(x)=cx$. Consider the $U$ the vector subspace generated by $A^i(x), i=0,...,m-1$. The restriction of $A^m$ to $U$ is $cI$. It implies that the minimal polynomial of this restriction divides $X^m-c$.