It is a well-known fact that commuting matrices over share a common eigenvector. This fact is a special case of the following result.

**Problem.** Let Prove that if for some then and share a common eigenvector.

**Solution.** The case is well-known. Suppose that Let be any eigenvalue of Let

Let be a non-zero element of such that and put

Then Therefore, by the maximality of we have and so is an eigenvector of both and

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