Making matrices better: Geometry and topology of polar and singular value decomposition

TL;DR

This paper explores the geometric and topological structure of matrix spaces, focusing on ranks, neighbors, and decompositions like polar and singular value decompositions, to better understand their properties and applications.

Contribution

It provides a novel geometric and topological analysis of matrix spaces, emphasizing the relationships between matrices of different ranks and their decompositions.

Findings

Characterization of matrix space topology and geometry

Insights into nearest orthogonal and singular neighbors

Applications to matrix decompositions

Abstract

Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal neighbor and nearest singular neighbor of a given matrix, both of which play central roles in matrix decompositions, and then against this visual backdrop examine the polar and singular value decompositions and some of their applications.