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I have the following problem:

Let $A$ and $B$ be two $n \times n$ matrices over $\mathbb{C}$ such that $AB = BA$ and $A^n = B^n = I$. Prove that $A$ and $B$ are simultaneously diagonalizable.

My thoughts: I know that for $A$ and $B$ to be simultaneously diagonalizable, I need to show that there exists an invertible matrix $C$ such that both $C^{-1}AC$ and $C^{-1}BC$ are diagonal. I am thinking that I can find/construct such a matrix from the condition that $A^n = B^n = I$, but I need further hints/direction to help guide me on this proof.

  • See here for instance: https://math.stackexchange.com/questions/236212/prove-that-simultaneously-diagonalizable-matrices-commute – kieransquared Jan 05 '23 at 01:55
  • @kieransquared Well you need to show that $A$ and $B$ are both diagonalizable in the first place (using the fact that $A^n = B^n = I$). – Ben Grossmann Jan 05 '23 at 01:56

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