hermitian matrix over C

• Feb 28th 2011, 08:59 PM
guin
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

Sorry for the mistake I made before in my question.
• Feb 28th 2011, 09:33 PM
topsquark
Quote:

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
• Mar 1st 2011, 12:25 AM
FernandoRevilla
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.