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Eigenvalues of Matrix Function

  1. Mar 2, 2013 #1
    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. jcsd
  3. Mar 2, 2013 #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?
     
  4. Mar 2, 2013 #3
    So you'd have T2v=t2v ... Tnv=tnv
     
  5. Mar 2, 2013 #4
    Then f(T)v=(f0+f1T +f2T2...)v = (f0+f1t1 +f2t2...)v =f(t1)v
     
  6. Mar 2, 2013 #5
    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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