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Orthogonal MAtrix

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Given the symmetric Matrix

    1 2
    2 5

    find an orthogonal matrix P such that C=BAB^t


    2. Relevant equations



    3. The attempt at a solution

    I found the eigenvalues to be [itex]3-(2\sqrt{2})[/itex] and [itex]3+(2\sqrt{2})[/itex]

    giving eigenvectors of
    [itex]
    [1,1-\sqrt{2}] [/itex] and [itex] [1,1+\sqrt{2}] [/itex]

    As the dot product of these vectors is 0 they are orthogonal.

    do I just normalise each vector and use them as the column vectors of P?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 27, 2009 #2
    It's going to be very difficult to make a statement about C=BAB^t in general. Knowing a symmetric matrix P, associated with an unlabeled matrix does very little to help.
     
  4. Oct 27, 2009 #3
    Sorry the matrix is

    A =

    1 2
    2 5

    find an orthogonal matrix P such that C=PAP^t where C is diagonal

    regards
     
  5. Oct 28, 2009 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Then find the eigenvectors of A. P will be an orthogonal matrix with the eigenvectors of A as rows.
     
  6. Oct 28, 2009 #5
    So I multiply P =
    [itex]
    \[ \left( \begin{array}{cc}
    1 & 1-\sqrt{2} \\
    1 & 1+\sqrt{2} \\
    \end{array} \right)\]
    [/itex]
    by A =
    [itex]
    \[ \left( \begin{array}{cc}
    1 & 2 \\
    2 & 5 \\
    \end{array} \right)\]
    [/itex]
    and PT =
    [itex]
    \[ \left( \begin{array}{cc}
    1 & 1 \\
    1-\sqrt{2} & 1+\sqrt{2} \\
    \end{array} \right)\]
    [/itex]

    which gives C =

    [itex]
    \[ \left( \begin{array}{cc}
    20-14\sqrt{2} & 0 \\
    0 & 14\sqrt{2}+20 \\
    \end{array} \right)\]
    [/itex]


    Is this right? I know that C is diagonal but isnt it supposed to have the eigenvalues on the main diagonal?
    regards
     
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