hermitian matrix over C

**guin** · February 28th 2011, 07:59 PM

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

Sorry for the mistake I made before in my question.

**topsquark** · February 28th 2011, 08:33 PM

Originally Posted by guin

Do anyone has an example of nxn hermitian matrix over complex which has n distinct eigenvalues?
If can make the n as small as possible. Thank you

If you want a simple one:
$\displaystyle \left [ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 5 \end{array} \right ]$

You can add window dressing to it, but it fits what you asked for. Did you want one less "trivial" to work on yourself?

-Dan

**FernandoRevilla** · February 28th 2011, 11:25 PM

Although the question has been already solved by topsquark , I'd like to add that $D=\textrm{diag}(\lambda_1,\ldots,\lambda_n)\in\mat hbb{C}^{n\times n}$ is hermitian iff $\lambda_i\in\mathbb{R}$ for all $i=1,\ldots,n$ . So, all diagonal real matrices with $\lambda_i\neq \lambda_j$ for all $i\neq j$ are hermitian and have $n$ distinct eigenvalues.

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