- #1
chocolatefrog
- 12
- 0
Homework Statement
αo, α1,..., αd [itex]\inℝ[/itex]. Show that αo + α1λ + α2λ2 + ... + αdλd [itex]\inℝ[/itex] is an eigenvalue of αoI + α1A + α2A2 + ... + αdAd [itex]\inℝ^{nxn}[/itex].
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) + ... + αd(Ad - Iλd)| = 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. :/