On partial orderings on the set of rectangular matrices (Q1890759)

Let \(\mathbb{C}_{m,n}\) and \(\mathbb{C}_n\) stand for the sets of \(m \times n\) and \(n \times n\) complex matrices respectively. The subset of \(\mathbb{C}_n\) consisting of Hermitian nonnegative definite matrices will be denoted by \(\mathbb{C}^\geq_n\), and its subset consisting of positive definite matrices by \(\mathbb{C}^>_n\). The authors introduce and investigate a new partial ordering on \(\mathbb{C}_{m,n}\). They give a characterization of this ordering and show that it collapses to the Loewner ordering when applied to \(C^\geq_n\). Moreover they give the comparison with the star ordering in general, and an alternative characterization in terms of the polar decomposition of matrices. Finally in the last section they consolidate the results of this paper in a tabular comparison.