The following question came up as a though while I was reading. I cannot see how to proceed on it.
Let us have $M_1,\ldots,M_n$ be commuting matrices. I know that that the generalized eigenspaces are the same across the $M_i$. However, is it true that for each generalized eigenspace, $V$, where $V$ is the generalized eigenspace of $\lambda_i$ for $M_i$, there exists $v\in V$ such that $v$ is an eigenvector for all the $M_i$.
A related question is if it is even true that the $M_i$ need have a common eigenvector (I see how an answer to this question implies an answer to my first question, so these seem equivalent)
I have tried to come up with examples for which what I said is not true, but this has been to no avail. From the generalized eigenspaces I do not see how to proceed.
Thank you for any help.