We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ⋆‐Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ⋆‐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.

Nonsingular systems of generalized Sylvester equations: An algorithmic approach

De Teran, Fernando;Iannazzo, Bruno;Robol, Leonardo
2019

Abstract

We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ⋆‐Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ⋆‐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.
2019
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/1452386
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