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Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1?

  1. Aug 2, 2011 #1
    1. The problem statement, all variables and given/known data
    If A is similar to A^(-1) (=inverse of A), must all the eigenvalues equal 1 or -1?


    2. Relevant equations



    3. The attempt at a solution

    I don't know why the textbook gives me the specific value 1 or -1.
    If A is similar to its inverse, are the eigenvalues really 1 or -1? Why? Help!
     
  2. jcsd
  3. Aug 2, 2011 #2

    jbunniii

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    Re: Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1

    Yes, the eigenvalues will be 1 or -1.

    First, if A and B are similar matrices, what can you say about their eigenvalues?

    Second, if [itex]\lambda[/itex] is an eigenvalue of A, what number must be an eigenvalue of A^(-1)?
     
  4. Aug 2, 2011 #3

    Dick

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    Re: Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1

    Suppose the eigenvalues are say, 2 and 1/2?
     
  5. Aug 2, 2011 #4

    jbunniii

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    Re: Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1

    Oops, you're right.
     
  6. Aug 2, 2011 #5

    Dick

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    Re: Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1

    Sanglee, I think the 1 and -1 are just to lead you into giving a wrong answer by thinking too quickly. It's true if A is a 1x1 matrix. Just think of a 2x2 matrix that is similar to its inverse without the diagonal entries being 1 or -1. Diagonal matrices will do.
     
    Last edited: Aug 2, 2011
  7. Aug 2, 2011 #6
    Re: Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1

    Oh~~~~~~~~I got it!!! it's really simple question, but I thought it was complicated. hehe

    So, A and inverse of A are similar, so their eigenvalues are same.
    if one of A's eigenvalues is n, a eigenvalues of its inverse will be 1/n.
    But the two matrices are similar, so n=1/n
    Then, n^2=1, so n=1or-1

    Is it right?
    Thanks guys!
     
  8. Aug 3, 2011 #7

    Dick

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    Re: Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1

    The two eigenvalues don't have to be equal. That's the mistake jbunniii made and the one the poser of the problem assumed you might make. Look at post #3.
     
  9. Aug 3, 2011 #8
    Re: Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1

    Oh, i understand it now :) Thanks!
     
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