If I consider two square matrices $A$ and $B$ such that $A B = B A =0$ and I know eigenvalues and eigenvectors of $A$, is it possible to get informations about the spectrum of $B$?
In particular, I have to diagonalize operators (I only know that they are matrices of a fixed dimension) of the form $ A + B $ and I know the spectrum of $A$.
EDIT: I also know that $A$ and $B$ have the same set of eigenvalues, excluding at most a finite set. The thing I really need is to find the eigenvectors in a general way, without looking explicitly at the form of B.
If $A$ and $B$ are Hermitian matrices with $AB = 0$, then you know that any eigenvector $x$ of $A$ associated with $\lambda \neq 0$ satisfies $Bx = 0$. That's about all you can say about eigenvalues.
As for eigenvectors: commuting Hermitian matrices share an orthonormal basis of eigenvectors. If there are no repeated eigenvectors, then any eigenvector of one is an eigenvector of the other.