2026/07/15 by Andrada Pojar
#math.RA
It is proved that if \mathbbF is a field of positive characteristic p, and if m and n are positive integers such that m≥2, and n≤ p≤ mn-1, for every n× n nonderogatory matrix A∈ \mathbbMn(\mathbbF), with trace in \k.1_ mathbbF| k∈ \0,1,…,p-1\\, there exist m idempotent matrices E1, E2,…, Em, and a nilpotent matrix N, such that A=E1+E2+…+Em+N, with Nk=0, where k=n if p∈ \nm-1,nm-2\, k=n-1 if n∈ \2,3\ or p=nm-3, otherwise k=max(2,1+\lfloor(n-1)/(r)\rfloor), if n is even, and k=max(3,1+\lfloor(n-1)/(r)\rfloor), if n is odd, where r:=\lfloor(nm-p)/(2)\rfloor. Moreover, for n>p, A is the sum of two idempotent matrices, and a square zero one, if n is even, and it is sum of two idempotent matrices and one which third power is zero, if n is odd.