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Math Help - How do I find a 2x2 diagonal matrix?

  1. #1
    Member zzizi's Avatar
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    How do I find a 2x2 diagonal matrix?

    Hi

    If a 2x2 matrix is:
    [1,2
    2,1]


    How do I find an invertible matrix P and diagonal matrix D such that : p^-1*A*P= D ?

    I would really appreciate help with this.
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  2. #2
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    Re: How do I find a 2x2 diagonal matrix?

    Hint: find the eigenvalues and eigenvectors of your matrix.
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  3. #3
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    Re: How do I find a 2x2 diagonal matrix?

    I found them, the eigenvalues are -1 and 3 and the eigenvectors are (1,1,1) and (1,1,-1). What do I do next?
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  4. #4
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    Re: How do I find a 2x2 diagonal matrix?

    Your eigenvectors are a bit long: they have three elements instead of 2. Once you have your eigenvectors and eigenvalues, you apply the spectral decomposition theorem. Spectral theorem - Wikipedia, the free encyclopedia . In the case of real matrices, "Hermitian" boils down to "symmetric".
    Sorry for being so cryptic, but once you have the eigenstructure, the answer is so easy that to say anything more would give it away.
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  5. #5
    Member zzizi's Avatar
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    Re: How do I find a 2x2 diagonal matrix?

    Sorry that was a typo, it should be (1,1) and (1,-1).

    I know it's easy, so please just give it away.

    I just wanted someone to show how they worked it out, because I have tried to multiply the inverse of the matrix with the original matrix and a matrix made with the eigenvectors. But for some reason my answer does not turn out to me the diagonal matrix. I just wanted to check if I have missed something.
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  6. #6
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    Re: How do I find a 2x2 diagonal matrix?

    So if the original matrix is A and the matrix of eigenvalues is D and the matrix of eigenvectors is P..

    what is AP and what is PD?

    AP=PD

    So P^-1 A P=P^-1 P D = D
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  7. #7
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    Re: How do I find a 2x2 diagonal matrix?

    The diagonal matrix has the eigenvalues in the diagonal, and P is made up of the eigenvectors. When you normalize the eigenvectors before putting them (for example as column vectors) into P, you get  P^T=P^{-1}. That follows from the fact that for this kind of matrix the eigenvectors are orthogonal.
    Last edited by bartmeijer; April 22nd 2012 at 07:19 AM.
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  8. #8
    Member zzizi's Avatar
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    Re: How do I find a 2x2 diagonal matrix?

    Thank you both for your help, I appreciate it.
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