On the Sum of Strictly k-zero Matrices

Little Hermie B. Monterde, Agnes T. Paras

Abstract


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


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