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Show that if A is invertible and diagonalizable,then A^−1 is

  1. Dec 10, 2012 #1
    Show that if A is invertible and diagonalizable,
    then A^−1 is diagonalizable. Find a 2 ×2 matrix
    that is not a diagonal matrix, is not invertible, but
    is diagonalizable.


    Alright, I am having some trouble with the first part.
    So far, I have this:
    If A is diagnolizable then
    A=PDP^-1 where P is the matrix who's columns are eigenvectors and D is the diagonal matrix of eigevenvalues of A.


    (A)^-1=(PDP^-1)^-1
    A^-1=PDP^-1

    How's that?
     
  2. jcsd
  3. Dec 10, 2012 #2

    Dick

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    Re: Diagonlizable

    Just because D is diagonal doesn't mean D=D^(-1). And after you've fixed that, how do you know D is invertible?
     
    Last edited: Dec 10, 2012
  4. Dec 10, 2012 #3
    Re: Diagonlizable

    Oh oops.
    So
    A^-1=P * D^-1 * P^-1

    hmm, does D have to be invertible?
    Can't you have eigen values of 0 and 2
    so D looks like this:
    0 0
    0 2
    which is not invertible?
     
  5. Dec 10, 2012 #4

    Dick

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    Re: Diagonlizable

    They told you A is invertible. Doesn't that mean D has to be invertible? Can you prove that?
     
  6. Dec 11, 2012 #5
    Re: Diagonlizable

    I'm confused on why you would have to prove that D is invertible and if it is always invertible.
    I'm calling D the diagonal matrix who's diagonal elements are the eigenvalues of A.
     
  7. Dec 11, 2012 #6

    Dick

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    Re: Diagonlizable

    D isn't invertible if it has a zero on the diagonal. But if it does then A has a zero eigenvalue and it's not invertible. I'm not sure whether you have to prove that or whether you can just say it. But it's not hard to prove.
     
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