On some characterizations of pairwise star orthogonality using rank and dagger additivity and subtractivity (Q1914248)

Let \(A_1, \dots, A_k\) be \(m\times n\) matrices with complex coefficients. Let \(A^\dagger\) denote the Moore-Penrose inverse of \(A\). The main result is that pairwise star orthogonality \(A^*_i A_k =0\) and \(A_i A^*_j =0\) for all \(i\neq j\) is equivalent to (i) \(A_i\leq_* \sum A_j\) and to (ii) \(A_i \leq_{rs} \sum A_j\) and \(\sum A^\dagger_j= (\sum A_j)^\dagger\), where \(\leq_*\) and \(\leq_{rs}\) denote, respectively, the star and rank-subtractivity (or minus) partial ordering. Further characterizations of the pairwise star orthogonality of \(k\) complex \(m\times n\) matrices are also presented.