Consider two $n\times n$ (hermitian) matrices $A$ and $B$. It is well know that they share a common eigenbasis if they commute, i.e. if $[A,B]=AB-BA=0$. I am interested in common eigenvectors in case $[A,B] \neq 0$.
The zero matrix has rank 0. It is tempting to try to read off the number of orthogonal common eigenvectors $m$ from the rank of the commutator as
$m = n - \text{rk}[A,B]$.
Is this statement true? If yes, where can I find a proof? If no, is there any other connection between the rank of a commutator and the eigenspaces of $A$ and $B$?
