Products of orthogonal projections and polar decompositions

TL;DR

This paper characterizes products of orthogonal projections and their polar decompositions, solving minimization problems and analyzing pseudoinverses for these products.

Contribution

It provides a comprehensive characterization of products of orthogonal projections and their algebraic properties, including polar decompositions and pseudoinverses.

Findings

Characterization of sets of products of orthogonal projections

Solutions to minimization problems involving these products

Determination of polar decompositions and Moore-Penrose pseudoinverses

Abstract

We characterize the sets $\XX$ of all products $PQ$, and $\YY$ of all products $PQP$, where $P,Q$ run over all orthogonal projections and we solve the problems $\arg\min\{\|P-Q\|: (P,Q) \in \cal Z\}$, for $\cal Z=\XX$ or $\YY.$ We also determine the polar decompositions and Moore-Penrose pseudoinverses of elements of $\XX.$