## Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues

## Problem 587

Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$.

Assume that $A-B$ is a nilpotent matrix.

Then prove that the eigenvalues of $A$ and $B$ are the same.

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