Im wondering if anyone can give me a good reference or answer this question which may have already be solved.
For a set of generic $n\times n$ matrices $A_1,A_2,...,A_k$, such that
- they share only a SINGLE eigenvector in common,
- the joint commutant of $A_1,A_2,...,A_k$ is trivial,
how can I guarantee there exists only this single one dimensional invariant subspace (corresponding to the span of that eigenvector)?
Googling around, it seems some progress has been made on saying whether invariant subspaces of dimension greater than 2 exist, but they rely on the fact that the matrices $A_1,A_2,...,A_k$ are chosen to have pairwise distinct eigenvalues.
I appreciate any help or direction! Thanks!