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Eigenvalue problem

  1. Mar 10, 2007 #1
    1. The problem statement, all variables and given/known data


    Two square matrices A and B of the same size do not commute.Prove that AB and BA has the same set of eigenvalues.

    I did in the following way:Please check if I am correct.
    Consider: det(AB-yI)*det(A) where y represents eigenvalues and
    I represents unit matrix
    =det[(AB-yI)A]
    =det[(AB)A-(yI)A]
    =det[A(BA)-A(yI)]
    =det(A)*det(BA-yI)
    det(A) is not equal to zero,in general.
    So,if det(AB-yI)=0,det(BA-yI)=0 also.
    hence, conclusion.
     
  2. jcsd
  3. Mar 10, 2007 #2

    Dick

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    Science Advisor
    Homework Helper

    That's basically it, but the argument is dubious. det(A) certainly could be zero. Try framing it this way. Let L be an eigenvalue of AB. Then ABx=Lx for some x. Act on both sides with B and conclude Bx is an eigenvector with eigenvalue L of BA. So if L is an eigenvalue of AB, it's a eigenvalue of BA.
     
  4. Mar 10, 2007 #3
    Oh!It's fantastic.I salute you whole-heartedly.
     
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