# Computing nearest stable matrix pairs

## TL;DR

This paper introduces a new approach to find the nearest stable matrix pair by reformulating the problem with dissipative Hamiltonian matrix pairs, enabling efficient computation of stable approximations.

## Contribution

The paper proposes a reformulation using dissipative Hamiltonian matrix pairs, simplifying the feasible set and allowing fast gradient methods for stability approximation.

## Key findings

- Reformulation with DH matrix pairs simplifies the feasible set.
- The method enables fast gradient-based computation of stable matrix pairs.
- The approach provides a practical solution for stability approximation in matrix pairs.

## Abstract

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair $(E,A)$, minimize the Frobenius norm of $(\Delta_E,\Delta_A)$ such that $(E+\Delta_E,A+\Delta_A)$ is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian (DH) matrix pairs: A matrix pair $(E,A)$ is DH if $A=(J-R)Q$ with skew-symmetric $J$, positive semidefinite $R$, and an invertible $Q$ such that $Q^TE$ is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.