# A Complex Matrix which commutes with No Nonzero Nilpotent matrices

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