Another equivalent characterization of EP matrices is that the range of A is orthogonal to the nullspace of A.
EP matrices were introduced in 1950 by Hans Schwerdtfeger,[1][3] and since then, many equivalent characterizations of EP matrices have been investigated through the literature.
[4] The meaning of the EP abbreviation stands originally for Equal Principal, but it is widely believed that it stands for Equal Projectors instead, since an equivalent characterization of EP matrices is based in terms of equality of the projectors AA+ and A+A.
When A is an EP matrix, the range of A is precisely perpendicular to the null-space of A.
Weakening the normality condition to EPness, a similar statement is still valid.