On sums of three square-zero matrices Original Research Article

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×n matrix A is a sum of three square-zero matrices, the dim ker(A−αI)less-than-or-equals, slant3n/4 for any scalar α≠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.