On some characterizations of pairwise star orthogonality using rank and dagger additivity and subtractivity Original Research Article

The main results are that pairwise star orthogonality, A*iAj = 0 and AiA*j = 0 for all i ≠ j, where A1, …, Ak are complex m × n matrices, is equivalent to (i) Ai less-than-or-equals, slant* ∑ Aj and to (ii) Ai less-than-or-equals, slantrs ∑ Aj and ∑ A†j = (∑ Aj)†, where i = 1, …, k; here the superscript † denotes the Moore-Penrose inverse, while less-than-or-equals, slant* and less-than-or-equals, slantrs denote, respectively, the star and rank-subtractivity (or minus) partial orderings. Five more characterizations of the pairwise star orthogonality of k complex m × n matrices are also presented. Our characterizations extend earlier results by Hartwig and Styan (1986) for k = 2.