Eigenvalues of a sum of power of a matrix

Question

$A$ 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?

Answer 1

HINT

If $A$ is diagonalizable, say $A = VDV^{-1}$ then $$ B = \sum_{k=1}^n \left(VDV^{-1}\right)^k = \sum_{k=1}^n VD^kV^{-1} = V \left(\sum_{k=1}^n D^k \right)V^{-1} $$ and $D$ is a diagonal matrix. Can you take it from here?

Answer 2

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 $\phi(x)\in\mathbb{R}[x]$

Answer 3

I want to add a bit with gt6989b that if you consider it in complex $\operatorname{GL}(n, \Bbb C)$ then it will always be equivalent with a Jordan canonical block form.

Now for your answer for any polynomial $p(x)$; $p(A)v=P(\lambda)v$ holds if $Av=\lambda v$.

Answer 4

Let $\lambda$ be an eigenvalue of $A$ and $v$ a corresponding eigenvector. Then $$ Bv = (A+A^2+\cdots + A^n)v\\ = Av+A^2v + \cdots +A^nv\\ = \lambda v + \lambda^2v + \cdots + \lambda^nv\\ = (\lambda + \lambda^2+\cdots + \lambda^n)v $$