Nonsingular systems of generalized Sylvester equations: an algorithmic approach

TL;DR

This paper develops a stable algorithm to determine the uniqueness of solutions for systems of generalized Sylvester equations, using spectral analysis and periodic Schur decomposition, with practical computational efficiency.

Contribution

It introduces a novel spectral characterization for nonsingularity and a backward stable algorithm for solving these systems efficiently.

Findings

Characterization of nonsingularity via spectral properties.

Development of an $O(n^3r)$ backward stable algorithm.

Reduction of complex systems to simpler periodic systems.

Abstract

We consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $\star$-Sylvester equations with $n\times n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized $\star$-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition, and leads to a backward stable $O(n^3r)$ algorithm for computing the (unique) solution.