Eigenvalues of Matrix Function

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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)


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.
 

Answers and Replies

  • #2
HallsofIvy
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Well, how about a direct calculation? Suppose v is an eigenvector of matrix T with eigenvalue [itex]t_1[/itex]. That is, [itex]Tv= t_1v[/itex]. Okay, so what is [itex]T^2v[/itex]? [itex]T^3v[/itex], etc?
 
  • #3
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So you'd have T2v=t2v ... Tnv=tnv
 
  • #4
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Then f(T)v=(f0+f1T +f2T2...)v = (f0+f1t1 +f2t2...)v =f(t1)v
 
  • #5
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Then for all eigenvalues... λ=tn f(T)v=f(tn)v , therefore f(B) has eigenvalues f(t1), f(t2),... f(tn)
 

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