# Eigenvalues of Matrix Function

## Homework Statement

Define a matrix function f(T) of an nxn matrix T by its Taylor series f(T)=f0 +f1T +f2T2+...
Show that if matrix T has the eigenvalues t1,t2...tn, then f(T) has eigenvalues f(t1), f(t2)...f(tn)

## The Attempt at a Solution

I am at a loss of how to prove this, could someone help me with this problem? I have no idea where to start.

HallsofIvy
Well, how about a direct calculation? Suppose v is an eigenvector of matrix T with eigenvalue $t_1$. That is, $Tv= t_1v$. Okay, so what is $T^2v$? $T^3v$, etc?