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Matrix diagonalization

  1. Mar 31, 2009 #1
    The question is "give an example of a square matrix A such that A^2 is diagonalizable but A is not."

    I know that if A^2 is diagonalizable, A^2 = P(D^2)P^-1. And if A is not diagonalizable, there is no invertible matrix P and diagonal matrix D such that A=PDP^-1.

    However I'm not sure how to begin finding an example.

    Can someone explain how to go about finding this?
     
  2. jcsd
  3. Apr 1, 2009 #2

    lanedance

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    Homework Helper

    I'm assuming you're talking about a matrix over the reals...

    The diagonalising matrix P is composed of eignevectors, so the eigenvectors need to exist and span dimension n, for the nxn matrix diagonlised form to exist

    so you could try and look for a matrix A where no real eigenvalues exist... but A^2 has real eigenvalues...

    2x2 is a good place to start...
     
  4. Apr 1, 2009 #3

    HallsofIvy

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    And i and -i are good eigenvalues to start with!
     
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