On the closest stable/unstable nonnegative matrix and related stability radii

TL;DR

This paper investigates the problem of finding the closest stable or unstable non-negative matrix to a given matrix, providing explicit solutions for unstable cases and an iterative method for stable cases, with analysis of complexity and numerical validation.

Contribution

It introduces an explicit solution for the closest unstable non-negative matrix and an iterative algorithm for the closest stable matrix, analyzing their properties and complexity.

Findings

Closest unstable matrix can be explicitly computed.

Iterative algorithm converges to a local minimum for stable matrices.

Number of local minima can grow exponentially with dimension.

Abstract

We consider the problem of computing the closest stable/unstable non-negative matrix to a given real matrix. This problem is important in the study of linear dynamical systems, numerical methods, etc. The distance between matrices is measured in the Frobenius norm. The problem is addressed for two types of stability: the Schur stability (the matrix is stable if its spectral radius is smaller than one) and the Hurwitz stability (the matrix is stable if its spectral abscissa is negative). We show that the closest unstable matrix can always be explicitly found. For the closest stable matrix, we present an iterative algorithm which converges to a local minimum with a linear rate. It is shown that the total number of local minima can be exponential in the dimension. Numerical results and the complexity estimates are presented.