The product of matrix subspaces

TL;DR

This paper investigates the geometric properties of matrix subspaces involved in matrix factorizations, focusing on curvature and the concept of factorizable subspaces, with applications to classical theorems.

Contribution

It introduces the notion of factorizable matrix subspaces and analyzes their geometric structure, extending classical results like the Craig-Sakamoto theorem.

Findings

Curvature of matrix subspace products is characterized.

The concept of factorizable matrix subspaces is developed.

Classical theorems are interpreted within this new framework.

Abstract

In factoring matrices into the product of two matrices operations are typically performed with elements restricted to matrix subspaces. Such modest structural assumptions are realistic, for example, in large scale computations. This paper is concerned with analyzing associated matrix geometries. Curvature of the product of two matrix subspaces is assessed. As an analogue of the internal Zappa-Sz\'ep product of a group, the notion of factorizable matrix subspace arises. Interpreted in this way, several classical instances are encompassed by this structure. The Craig-Sakamoto theorem fits naturally into this framework.