On sums of three square-zero matrices

Abstract

Wang and Wu characterized matrices which are sums of two square-zero matrices, and proved that every matrix with trace-zero is a sum of four square-zero matrices. Moreover, they gave necessary or sufficient conditions for a matrix to be a sum of three square-zero matrices. In particular, they proved that if an n x n matrix A is a sum of three square-zero matrices, the dim ker(A - alpha I) less than or equal to 3n/4 for any scalar alpha not equal 0. Proposition 1 shows that this condition is not necessarily sufficient for the matrix A to be a sum of three square-zero matrices, and characterizes sums of three square-zero matrices among matrices with minimal polynomials of degree 2. (C) 2000 Elsevier Science Inc, All rights reserved.