# Eigenvalues of Matrix Function

1. Mar 2, 2013

### digipony

1. The problem statement, all variables and given/known data
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)

2. Relevant equations

3. 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.

2. Mar 2, 2013

### HallsofIvy

Staff Emeritus
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?

3. Mar 2, 2013

### digipony

So you'd have T2v=t2v ... Tnv=tnv

4. Mar 2, 2013

### digipony

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

5. Mar 2, 2013

### digipony

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