Nonderogatory matrices as sums of idempotent and nilpotent matrices

Abstract

We prove that, over a field F of characteristic p>0, every nil-clean nonderogatory matrix A with tr(A)≠1 has a decomposition A=E+V such that E2=E and Vp+1=0. If tr(A)=1 then we have a similar decomposition with Vp+2=0. Consequently, every nilpotent matrix over F is a sum of an idempotent matrix and a nilpotent matrix of nilpotence index at most p+1.