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Find a unitary matrix U such that U*AU is diagonal

  1. May 10, 2012 #1
    For the following matrix A, find a unitary matrix U such that U*AU is diagonal:
    A =
    1 2 2 2
    2 1 2 2
    2 2 1 2
    2 2 2 1

    I found the eigenvalues to be -1,-1,-1,7
    and the eigenvectors to be (v1)=(-1,1,0,0),(v2)=(-1,0,1,0),(v3)=(-1,0,0,1),(v4)=(1,1,1,1)
    Normalize these vectors: ||(v1)||=sqrt(2),||(v2)||=sqrt(2), ||(v3)||=sqrt(2), ||(v4)||=2
    So a unitary matrix is
    U=
    1/sqrt(2) -1/sqrt(2) -1/sqrt(2) -1/sqrt(2)
    1/sqrt(2) 0 0 1/2
    0 1/sqrt(2) 0 1/2
    0 0 1/sqrt(2) 1/2

    But this does not satisfy U*AU is diagonal, so I'm thinking I want to change the order of the vectors. But how do I know which one is satisfies the condition? (trial and error is rather tedious)
     
  2. jcsd
  3. May 10, 2012 #2

    sharks

    User Avatar
    Gold Member

    U is simply the columns of eigenvectors arranged one next to the other, in the same order as the eigenvalues for the diagonal matrix.

    For example, if the diagonal matrix:
    [tex]D=U^{-1}AU=
    \displaystyle\left[ {\begin{array}{*{20}{c}}
    -1&0&0&0 \\
    0&-1&0&0 \\
    0&0&-1&0 \\
    0&0&0&7
    \end{array}} \right][/tex]
    Then, the corresponding matrix U will have the first column as the eigenvector corresponding the eigenvalue, [itex]\lambda_1 = -1[/itex], and so on, with the last column of matrix U as the eigenvector corresponding to [itex]\lambda_4 = 7[/itex].
     
    Last edited: May 10, 2012
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