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Let A be a $n\times n$ complex matrix. We need to prove that A has n distinct eigen values in $\mathbb{C}$ iff A commutes with no non-zero Nilpotent matrix.I am not getting any hint how to proceed, shall be pleased for your comments

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As you probably know, in order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are so far; this will prevent people from telling you things you already know, and help them write their answers at an appropriate level. Also, many find the use of imperative ("Find", "Show") to be rude when asking for help; please consider rewriting your post. – Arturo Magidin Apr 16 '12 at 19:34
the matrix you are looking for is $\prod_{\text{unique eigenvalues}} (A - \lambda_i)$ which is non zero and nilpotent iff A fails to have distinct eigenvalues – mike Apr 16 '12 at 19:35
up vote 2 down vote accepted

Hint: if $A$ commutes with the nilpotent matrix $N$ and $Av = \lambda v$, then $ANv = \lambda N v$. Show that $Nv = 0$.

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