I have a square matrix A, that is nilpotent. I want to show that the eigenvalue, say L = 0.

I'm not sure how to approach this problem because I have used the definition of an eigenvalue and eigenvector of A corresponding to L:

AX = LX

But I cannot use the fact that A is nilpotent because I don't know how to prove that (A^m)X = (L^m)X. Can anybody help with this?

Thank you!