An nxn matrix is said to be nilpotent if for some positive k. Show that all eigenvalues of a nilpotent matrix are 0.
I have proved by math induction that, for , is an eigenvalue of .
I don't know if that should help.
In particular, if an eigenvalue of A, then is an eigenvalue of - that is, for some non-zero vector v, . But for any vector so we have , with v non-zero.