# Can anyone explain why commuting matrices share common eigenvector? [duplicate]

If two matrices $A$ and $B$ commute with each other, why would they share some eigenvector?

Does that mean that an eigenvector for $A$ is also an eigenvector for $B$ and vice-versa?

However, when the underlying field is not algebraically closed, the answer to your question is clearly no. E.g. consider any two $2\times2$ real rotation matrices for any two different angles other than $0$ or $\pi$. They simply have no eigenvalues or eigenvectors over $\mathbb R$.