Uniqueness of solution of systems of generalized Sylvester and conjugate-Sylvester equations
Fernando De Terán, Bruno Iannazzo
TL;DR
This work provides a spectral, pencil-based characterization for the unique solvability of periodic systems built from generalized Sylvester and conjugate-Sylvester equations with equal-sized unknowns and square coefficients. By reducing any general system to irreducible periodic instances (with at most one conjugate equation), the authors derive precise conditions: in the purely generalized-Sylvester case, two specific matrix pencils must be regular and have disjoint spectra; when exactly one conjugate-Sylvester equation is present, augmented pencils incorporating the conjugate-terms must likewise be regular and spectrally disjoint. The paper also addresses sign conventions in the second term, offers alternative characterizations via block permutations, and corrects a prior result (Dipr19) with counterexamples, while extending the framework to rectangular unknowns. These results yield practical, verifiable criteria for uniqueness and outline future work on varying unknown sizes and non-square coefficients, advancing both theory and numerical treatment of Sylvester-type systems.
Abstract
We provide a characterization for a periodic system of generalized Sylvester and conjugate-Sylvester equations, with at most one generalized conjugate-Sylvester equation, to have a unique solution when all coefficient matrices are square and all unknown matrices of the system have the same size. We also present a procedure to reduce an arbitrary system of generalized Sylvester and conjugate-Sylvester equations to periodic systems having at most one generalized conjugate-Sylvester equation. Therefore, the obtained characterization for the uniqueness of solution of periodic systems provides a characterization for general systems of generalized Sylvester and conjugate-Sylvester equations.