Hi everyone,

Ive been looking at the following problem, but its not entirely in my area and some potential solutions seem to rely on algebraic geometry. Maybe thats just a complicated way to solve the problem.

Im wondering if anyone can give me a good reference or answer to this question

For any set of $n×n$ matrices $A_1,A_2,...,A_k$, such that

1. they share only a SINGLE eigenvector in common
2. the joint commutant of $A_1,A_2,...,A_k$ and $A^*_1,A^*_2,...,A^*_k$ 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.

Thanks for the help!

edit: changed some wording so I dont mislead anyone.