What are the eigenvalues of P(A)?

[chocolatefrog]

Homework Statement

α_o, α₁,..., α_d $\inℝ$. Show that α_o + α₁λ + α₂λ² + ... + α_dλ^d $\inℝ$ is an eigenvalue of α_oI + α₁A + α₂A² + ... + α_dA^d $\inℝ^{nxn}$.


2. The attempt at a solution

If λ is an eigenvalue of A, then |A - Iλ| = 0. Also, λⁿ is an eigenvalue Aⁿ. So we basically have to somehow prove the following equation (after rearranging):

|α₁(A - Iλ) + α₂(A² - Iλ²) + ... + α_d(A^d - Iλ^d)| = O

3. Relevant equations

I can't seem to get my head around this one. I almost used the triangular inequality to prove it before I realized that these are determinants we are dealing with, not absolute values. :/

 

[micromass]

So let $P(z)$ be a polynomial. We wish to prove that $P(\lambda)$ is an eigenvalue of $P(A)$.

For each $\mu\in \mathbb{C}$, we can write

$$P(z)-\mu=r_0(z-r_1)...(z-r_n)$$

Thus

$$P(A)-\mu I = r_0(A-r_1 I)...(z-r_n I)$$

So $P(A)-\mu I$ is invertible iff all $(A-r_i I)$ is invertible. What does that imply for the eigenvalues?