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Nonsingular systems of generalized Sylvester equations: An algorithmic approach
Author(s) -
De Terán Fernando,
Iannazzo Bruno,
Poloni Federico,
Robol Leonardo
Publication year - 2019
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2261
Subject(s) - sylvester equation , sylvester matrix , invertible matrix , mathematics , uniqueness , matrix (chemical analysis) , sylvester's law of inertia , decomposition , pure mathematics , algebra over a field , mathematical analysis , eigenvalues and eigenvectors , symmetric matrix , polynomial matrix , ecology , physics , materials science , matrix polynomial , quantum mechanics , polynomial , composite material , biology
Summary 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 3 r ) algorithm for computing the (unique) solution.