Global product structure for a space of special matrices

TL;DR

This paper explores the topological structure of spaces of Hurwitz symmetric and Metzler matrices, revealing their manifold properties and analyzing robustness, with applications to biological models like insulin regulation.

Contribution

It establishes a product manifold structure for Hurwitz symmetric matrices and analyzes the robustness of Hurwitz Metzler matrices using differential topology.

Findings

Space of Hurwitz symmetric matrices forms a product manifold.

The structure of Hurwitz Metzler matrices allows robustness analysis.

Application to insulin model demonstrates practical relevance.

Abstract

The importance of the Hurwitz Metzler matrices and the Hurwitz symmetric matrices can be appreciated in different applications: communication networks, biology and economics are some of them. In this paper, we use an approach of differential topology for studying such matrices. Our results are as follows: the space of the $n\times n$ Hurwitz symmetric matrices has a product manifold structure given by the space of the $(n-1) \times (n-1)$ Hurwitz symmetric matrices and the euclidean space. Additionally we study the space of Hurwitz Metzler matrices and these ideas let us do an analysis of robustness of Hurwitz Metzler matrices. In particular, we study the Insulin Model as application.