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Math Help - hermitian matrix over C

  1. #1
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    hermitian matrix over C

    Do anyone has an example of nxn hermitian matrix with complex entries which has repeated eigenvalues?
    If can make the n as small as possible. Thank you
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by guin View Post
    Do anyone has an example of nxn hermitian matrix with complex entries which has repeated eigenvalues?
    If can make the n as small as possible. Thank you
    How about I_2?
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    But if I would like to have entries such as 2+i or other which will involve the term i ?
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    The matrix \begin{bmatrix}1&i&0\\ -i&1&0\\ 0&0&0\end{bmatrix} has a repeated eigenvalue 0.

    Edit. Or if you want each eigenvalue to be repeated then you'll need a 4x4 matrix:

    \begin{bmatrix}1&i&0&0\\ -i&1&0&0\\ 0&0&1&i\\ 0&0&-i&1\end{bmatrix}.
    Last edited by Opalg; March 4th 2011 at 05:42 AM.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by guin View Post
    But if I would like to have entries such as 2+i or other which will involve the term i ?

    If A\in\mathbb{C}^{n\times n} is hermitian all its eigenvalues belong to \mathbb{R} and is diagonalizable.

    If you mean that A has only one repeated eigenvalue \lambda (therefore n\geq 2) then, A is similar to D=\lambda I_n that is, there exists P\in\mathbb{C}^{n\times n} non singular such that P^{-1}AP=\lambda I_n or equivalently A=\lambda I_n so, A has to be a real and scalar matrix.
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