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.

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- April 18th 2010, 08:14 AMdwsmithNilpotent
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. - April 18th 2010, 08:22 AMHallsofIvy
Yes, that certainly does help!(Clapping) I presume that in doing that you also showed that if v is an eigenvector of A corresponding to eigenvalue then v is also an eigenvector of corresponding to eigenvalue .

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. - April 18th 2010, 08:29 AMdwsmith
- April 18th 2010, 08:59 AMHallsofIvy
Well, what do you conclude

**from**? - April 18th 2010, 09:11 AMdwsmith
I should probably conclude lambda is zero.