The factorization of matrices into products of positive definite factors

TL;DR

This paper explores the factorization of matrices into positive-definite factors, linking it to optimal transportation and spectral control, with a focus on minimal factors and conditioning trade-offs.

Contribution

It introduces a novel analysis of positive-definite matrix factorization via Monge-Kantorovich transportation, providing computational methods to determine minimal factors and spectral properties.

Findings

Determines the minimal number of positive-definite factors for spectral control.

Links matrix factorization to optimal transportation on Gaussian distributions.

Provides computational techniques for conditioning and factor trade-offs.

Abstract

Positive-definite matrices materialize as state transition matrices of linear time-invariant gradient flows, and the composition of such materializes as the state transition after successive steps where the driving potential is suitably adjusted. Thus, factoring an arbitrary matrix (with positive determinant) into a product of positive-definite ones provides the needed schedule for a time-varying potential to have a desired effect. The present work provides a detailed analysis of this factorization problem by lifting it into a sequence of Monge-Kantorovich transportation steps on Gaussian distributions and studying the induced holonomy of the optimal transportation problem. From this vantage point we determine the minimal number of positive-definite factors that have a desired effect on the spectrum of the product, e.g., ensure specified eigenvalues or being a rotation matrix. Our approach is computational and allows to identify the needed number of factors as well as trade off their conditioning number with their actual number.