Probably, this question has already been answered, but I did not find an answer.
If a matrix $A$ is hermitian and an eigenvalue $\lambda$ has multiplicity $k$, are there always $k$ pairwise orthogonal vectors $x_1,...,x_k$ with $Ax_i=\lambda x_i$ for all $i=1,...,k$ ? If yes, how can this be proven ?
I know that a hermitian matrix has real eigenvalues, and that the eigenvectors to different eigenvalues have to be orthogonal. But I am not quite sure about the situation for multiple eigenvalues.