Eigenvalues of a sum of power of a matrix $A$ is defined as a real $n×n$ matrix. $B$ is defined as:
$$B=A+A^2+A^3+A^4+ \dots +A^n$$
What's the relation between eigenvalues of $A$ and eigenvalues of $B$?
Can anyone give me some materials?

Try to prove that if $\lambda$ is an eigenvalue of $A$ then $\phi(\lambda)$ is an eigenvalue of $\phi(A)$ for every $A\in \mathbb{R}^{n\times n}$ and $\phi(x)\in\mathbb{R}[x]$
