# Thread: hermitian matrix over C

1. ## 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.

2. 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:
$\displaystyle \displaystyle \left [ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 5 \end{array} \right ]$

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

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