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Find orthogonal P and diagonal matrix D

  1. Jun 6, 2010 #1
    1. The problem statement, all variables and given/known data

    A= [1 -1 0]
    [-1 2 -1]
    [0 -1 1]
    find orthogonal matrix P and diagonal matrix D such that P' A P = D

    2. Relevant equations

    3. The attempt at a solution
    i got eigenvalues are 0, 1, 3 which make D=[0 0 0; 0 1 0; 0 0 3]
    how to find P. because in my solution they mentioned about normalised eigenvectors.
  2. jcsd
  3. Jun 6, 2010 #2


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    Find the eigenvectors. The columns of P are the eigenvectors.
  4. Jun 6, 2010 #3
    i know after i got eigenvalues, i can find eigenvectors which is P.
    my question is that in my solution, P are normalised eigenvectors. why they use normalised eigenvector instead of the eigenvector?
  5. Jun 6, 2010 #4

    D H

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    If you used unnormalized eigenvectors the diagonalization equation is P-1AP=D instead of P*AP=D. The inverse is particularly easy to find with normalized eigenvectors. If P is constructed from normalized, orthogonal eigenvectors then P will be an orthogonal matrix, making P-1=P*.
  6. Jun 6, 2010 #5
    thanks, got it
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