Eigenvalues of Matrix Function

[digipony]

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...t_n, then f(T) has eigenvalues f(t₁), f(t₂)...f(t_n)

Homework Equations

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?

 

[digipony]

So you'd have T²v=t²v ... Tⁿv=tⁿv

 

[digipony]

Then f(T)v=(f₀+f₁T +f₂T²...)v = (f₀+f₁t1 +f₂t²...)v =f(t₁)v

 

[digipony]

Then for all eigenvalues... λ=t_n f(T)v=f(t_n)v , therefore f(B) has eigenvalues f(t₁), f(t₂),... f(t_n)