Partial orderings, preorderings, and the polar decomposition of matrices (Q1300828)

The starting point is the well known polar decomposition of a complex \(m\times n\)-matrix. It can be made unique by imposing additional conditions on the factors. Each partial order on one of the domains of this decomposition gives rise to a preordering on \({\mathbb C}^{m\times n}\). If we are given a partial order on both domains, then the intersection of the corresponding preorderings is a partial ordering on \({\mathbb C}^{m\times n}\). The domains of the unique polar decomposition are now endowed, respectively, with one of three partial orderings (Löwner, star, and rank subtractivity), and the corresponding preorderings and partial orderings are investigated.