## All the Eigenvectors of a Matrix Are Eigenvectors of Another Matrix

## Problem 51

Let $A$ and $B$ be an $n \times n$ matrices.

Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$.

Then prove that each eigenvector of $A$ is an eigenvector of $B$.

(It could be that each eigenvector is an eigenvector for distinct eigenvalues.)

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