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Math Help - Orthogonally diagonalize the matrix

  1. #1
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    Orthogonally diagonalize the matrix

    How do you orthogonally diagonalize the following matrix:

     <br />
A=\begin{pmatrix}1&0&0\\0&3&-2\\0&-2&3\end{pmatrix}<br />
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  2. #2
    Senior Member Twig's Avatar
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    hi

    hi

    You need to find the eigenvalues of the matrix, and find the corresponding eigenvectors. Do you know how to do this?

     A = PDP^{T} , since A is orthogonally diagonizable, P will be an orthogonal matrix, hence  P^{-1}=P^{T} .

    An easy way to find the eigenvalues of this matrix would be a cofactor expansion for example, or you could use the fact that it is block triangular.

    Itīs quite a long process to describe if you have no idea whatsoever on how to find eigenvalues or eigenvectors, and I would in that case suggest you start there.
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  3. #3
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    I think the eigenvalues are 5, 1 and 1 or just 5 and 1? and the eigenvectors are <0 -1 1>, <0 1 1> and <1 0 0>?
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  4. #4
    Senior Member Twig's Avatar
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    hi

    hi

    Yes, the eigenvalus are  \lambda_{1} = 5 \; \lambda_{2} = 1 \; \lambda_{3} = 1 so the multiplicity of eigenvalue 1 is two. This means that the dimension of the nullspace of the matrix  (A- 1 \cdot I) has dimension 2.

    The eigenvectors you wrote are correct.

    There is a theorem that says that eigenvectors from different eigenspaces to a symmetric matrix are orthogonal, so you only need to verify that the two vectors corresponding to eigenvalue 1 are orthogonal to each other.
    Which they are here.

    Now you normalize your eigenvectors and put them in a matrix P, and the eigenvalues you put in a diagonal matrix D, with the eigenvalue corresponding to the first column vector in the first column of D etc. Order matters here!

    And you are done!
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  5. #5
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    My P matrix is a little ugly but I'm hoping it is correct.

     <br />
\begin{pmatrix}0&1&0\\-\sqrt2/2&0&\sqrt2/2\\\sqrt2/2&0&\sqrt2/2\end{pmatrix}<br />
    Last edited by antman; April 21st 2009 at 03:45 PM.
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  6. #6
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    and my final would be the eigenvectors on the diagonal like \begin{pmatrix}5&0&0\\0&1&0\\0&0&1\end{pmatrix}?
    Last edited by antman; April 21st 2009 at 03:45 PM.
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  7. #7
    Senior Member Twig's Avatar
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    hi

    hi

    Yes, your matrix P is correct, and so is your matrix D.

     A = PDP^{T}

    I assumed you meant eigenVALUES on the diagonal, not eigenvectors

    good job
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