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Math Help - A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

  1. #1
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    A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

    Let A be a symmetric matrix over the real numbers. A^10 = I. Prove A^2 = I.

    I had this question in an exam but got 0/6 for it, so I'm not going to show my working here which was just wrong. How should I go about solving this question.

    Thanks
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    Super Member girdav's Avatar
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    Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

    Hint: A symmetric matrix over \mathbb{R} is diagonalizable, and the eigenvalues are real. What about the eigenvalues of A?
    Last edited by girdav; July 17th 2011 at 08:18 AM.
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    Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

    Quote Originally Posted by girdav View Post
    Hint: A symmetric matrix over \mathbb{R} is diagonalizable, and the eigenvalue are real. What about the eigenvalues of A?
    The diagonal of A is made of eigenvalues and A^10 = I, so A's eigenvalues are from the set {1,-1}. So A^2 =1.
    Thanks!

    But how do I know that a symmetric matrix is diagonalizable?
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    Super Member girdav's Avatar
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    Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

    It's true for symmetric real matrices. It's a classical result, which can be shown for example by induction on dimension of the vector space. You can find a proof here.
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    Re: A is symmetric matrix and over R. A^10 = I. Prove A^2 = I

    Ah! Thanks. That's what the real spectral theorem tells me!
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