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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?

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Although this question is different from "Do commuting matrices share the same eigenvectors?" (the linked question asks whether two commuting matrices have all eigenvectors in common, while the current question merely asks whether two commuting matrices have some eigenvectors in common), the answer by Algebraic Pavel does address your question if the field is complex. The same reasoning goes if the field is algebraically closed.

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 $\mathbb R$.

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