Linear maps on $M_n(\mathbb{R})$ preserving Schur stable matrices

TL;DR

This paper characterizes linear maps on real matrices that preserve Schur stable matrices, showing such maps have spectral radius at most one and fully describing those that preserve the entire set of Schur stable matrices.

Contribution

It provides a complete characterization of linear maps on $M_n({R})$ that preserve Schur stability, including spectral radius preservation conditions.

Findings

Spectral radius of stability-preserving maps is at most 1.

Maps that preserve all Schur stable matrices have spectral radius exactly 1.

The paper characterizes these maps explicitly on $M_n({R})$ and on the set of Schur stable matrices.

Abstract

An $n \times n$ matrix $A$ with real entries is said to be Schur stable if all the eigenvalues of $A$ are inside the open unit disc. We investigate the structure of linear maps on $M_n(\mathbb{R})$ that preserve the collection $\mathcal{S}$ of Schur stable matrices. We prove that if $L$ is a linear map such that $L(\mathcal{S}) \subseteq \mathcal{S}$, then $\rho(L)$ (the spectral radius of $L$) is at most $1$ and when $L(\mathcal{S}) = \mathcal{S}$, we have $\rho(L) = 1$. In the latter case, the map $L$ preserves the spectral radius function and using this, we characterize such maps on both $M_n(\mathbb{R})$ as well as on $\mathcal{S}^n$.