On the Sum of Strictly k -zero Matrices - Monterde, Little Hermie B.; Paras, Agnes T. (Science Diliman: A Journal of Pure and Applied Sciences; Vol 29 No 1 (2017), 2017)
Let k be an integer such that k ≥ 2. An n -by- n matrix A is said to be strictly k -zero if Ak = 0 and Am ≠ 0 for all positive integers m with m < k . Suppose A is an n -by- n matrix over a field with at least three elements. We show that, if A is a nonscalar matrix with zero trace, then (i) A is a sum of four strictly k -zero matrices for all k ∈{2,..., n}; and (ii) A is a sum of three strictly k -zero matrices for some k ∈{2,..., n }. We prove that, if A is a scalar matrix with zero trace, then A is a sum of five strictly k -zero matrices for all k ∈{2,..., n }. We also determine the least positive integer m , such that every square complex matrix A with zero trace is a sum of m strictly k -zero matrices for all k ∈{2,..., n }. Keywords: Nilpotent matrix, trace, Jordan canonical form