Partial orderings, preorderings, and the polar decomposition of matrices Original Research Article

When a unique decomposition of a complex rectangular matrix into two components is given, then a partial ordering on the domain of one of the components induces a preordering on the whole set of matrices, and partial orderings on the domains of both components induce a partial ordering on the whole set. In this not we consider the three main partial orderings (Lo¨wner, star, and rank subtractivity) on the respective domains of the two components of matrices subjected to polar decomposition (A. Ben-Israel. T.N.E. Greville, Generalized Inverses: Theory and Applications, Wiley, New York, 1974, p. 255) and investigate the resulting pre- and partial orderings.