Partial orderings, preorderings, and the polar decomposition of matrices

When a unique decomposition of a complex rectangular matrix into two compon ents is given, then a partial ordering on the domain of one of the componen ts induces a preordering on the whole set of matrices, and partial ordering s on the domains of both components induce a partial ordering on the whole set. In this note we consider the three main partial orderings (Lowner, sta r, and rank subtractivity) on the respective domains of the two components of matrices subjected to polar decomposition (A. Ben-Israel, T.N.E. Grevill e, Generalized Inverses: Theory and Applications, Wiley, New York, 1974, p. 255) and investigate the resulting pre- and partial orderings. (C) 1999 El sevier Science Inc. All rights reserved.