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The factorization of matrices into products of positive definite factors

Mahmoud Abdelgalil, Tryphon T. Georgiou

TL;DR

The paper addresses the problem of expressing a real matrix with positive determinant as a product of positive-definite factors by embedding the factorization into a geometric framework of gradient flows on Gaussian distributions and exploring the holonomy of Monge–Kantorovich transport. The authors develop a constructive method: first, analyze holonomy in the 2D Gaussian setting to obtain explicit bounds and formulas for the angular displacement, and then use these results to factor 2×2 matrices into 2–5 positive factors; they extend the approach to $\mathbb{R}^n$ by decomposing orthogonal blocks into 2×2 rotations, achieving a factorization into at most six symmetric factors, with Ballantine’s five-factor bound providing a theoretical limit. The work combines optimal transport, spectral control, and gradient-flow control to determine not only the minimal number of factors but also how to choose them to balance conditioning and factor count, offering a constructive route to Ballantine factorizations. Practically, this framework informs design of control protocols for ensembles of gradient-flow dynamics and establishes a principled link between holonomy in Gaussian transport and matrix spectral manipulation.

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.

The factorization of matrices into products of positive definite factors

TL;DR

The paper addresses the problem of expressing a real matrix with positive determinant as a product of positive-definite factors by embedding the factorization into a geometric framework of gradient flows on Gaussian distributions and exploring the holonomy of Monge–Kantorovich transport. The authors develop a constructive method: first, analyze holonomy in the 2D Gaussian setting to obtain explicit bounds and formulas for the angular displacement, and then use these results to factor 2×2 matrices into 2–5 positive factors; they extend the approach to by decomposing orthogonal blocks into 2×2 rotations, achieving a factorization into at most six symmetric factors, with Ballantine’s five-factor bound providing a theoretical limit. The work combines optimal transport, spectral control, and gradient-flow control to determine not only the minimal number of factors but also how to choose them to balance conditioning and factor count, offering a constructive route to Ballantine factorizations. Practically, this framework informs design of control protocols for ensembles of gradient-flow dynamics and establishes a principled link between holonomy in Gaussian transport and matrix spectral manipulation.

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.

Paper Structure

This paper contains 10 sections, 3 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Holonomy in the transport of Gaussian distributions
  3. Optimal Transport
  4. Holonomy of transport
  5. Angular displacement of segments
  6. Rotational dispacement in $\mathbb R^2$
  7. Holonomy of Gaussian triangles in $\mathbb R^2$
  8. Factorization into positive factors
  9. Factorization of rotation matrices in $\mathbb R^n$
  10. Concluding remarks

Key Result

Proposition 1

$\pi:\text{GL}^+(n)\rightarrow\text{Sym}^+(n)$ is a principal $\text{SO}(n)$-bundle.

Figures (1)

  • Figure 1: An illustration of the transitions $v\mapsto \Sigma_1^{1/2}v\mapsto \Phi_{21}\Sigma_1^{1/2}v\mapsto \Sigma_2^{-1/2}\Phi_{21}\Sigma_1^{1/2}v$.

Theorems & Definitions (4)

  • Proposition 1
  • Lemma 1
  • Theorem 1
  • Remark 1