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Two $n × n$ matrices $A$ and $B$ are said to be simultaneously diagonalizable if there exists an invertible $n × n$ matrix $S$ such that $S^{−1}AS$ and $S^{−1}BS$ are both diagonal.

How do I prove that if $A$ and $B$ are $n × n$ matrices such that $A$ has $n$ distinct eigenvalues and $AB = BA$, then $A$ and $B$ are simultaneously diagonalizable.

I am totally lost on this problem.

Jeansandcofffeee
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  • look here http://math.stackexchange.com/questions/1738173/how-do-i-prove-that-ab-ba/1738181#1738181 – Bérénice Apr 11 '16 at 23:06
  • Search questions for "simultaneously diagonalizable proof"; they will have compatible eigenspaces, so just find the eigenvectors for both. – Peter Diehr Apr 11 '16 at 23:10

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