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Math Help - Confused about eigenvectors of this hermitian matrix... due tomrrow morning!

  1. #1
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    Question Confused about eigenvectors of this hermitian matrix... due tomrrow morning!

    I'm being asked to find the spectral decomposition of a hermitian matrix... and I know that if i have 2 distinct eigenvalues, their eigenvectors MUST be normal to each other... thus... I am confused... becasue for this simple matrix...

    [1......3+4i]
    [3-4i......1]

    I am getting the distinct eigenvalues 6 and -4.... and

    eigenvectors [(3+4i)/5 1] and [-(3+4i)/5 1]

    which are definatley not orthogonal...

    what in blazes am I doing wrong here

    going crrrrrrrrrrazy!
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by douber View Post
    I'm being asked to find the spectral decomposition of a hermitian matrix... and I know that if i have 2 distinct eigenvalues, their eigenvectors MUST be normal to each other... thus... I am confused... becasue for this simple matrix...

    [1......3+4i]
    [3-4i......1]

    I am getting the distinct eigenvalues 6 and -4.... and

    eigenvectors [(3+4i)/5 1] and [-(3+4i)/5 1]

    which are definatley not orthogonal...

    what in blazes am I doing wrong here

    going crrrrrrrrrrazy!
    Those vectors are orthogonal! Remember that when the scalars are complex, you have to take the complex conjugates of the coefficients of the second vector when forming an inner product. So the inner product of the vectors is \tfrac{3+4i}5\Bigl(-\tfrac{3{\color{red}-}4i}5\Bigr) + 1. You'll find that this is zero.
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  3. #3
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    wow i feel stupid now lol... i forgot about that.. duh!

    thanks
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