@article{Shemesh1984CommonEO,
title={Common eigenvectors of two matrices},
author={D. Shemesh},
journal={Linear Algebra and its Applications},
year={1984},
volume={62},
pages={11-18}
}

Abstract A computable criterion is given for two square matrices to possess a common eigenvector, as well as a criterion for one matrix to have an eigenvector lying in a given subspace. Some applications are discussed.