I'm stuck to prove Nilpotent Matrix

[crazygrey]

Hi all,
If a square matrix A of dimension n*n is a nilpotent matrix,i.e, A^k=0 for k>=m iff A has eigenvalues 0 with multiplicity n and index m or less. I did prove by induction if A^k=0 then all the eigenvalues are zero. I'm lost when I want to prove the oppesite, i.e, if all eigenvalues are zero then A^k=0? Please help

 

[matt grime]

Put it in jordan normal form and it all drops out. Alternatively just think about the characteristic poly.