Nonsingular systems of generalized Sylvester equations: An algorithmic approach Articles uri icon

publication date

  • October 2019

start page

  • 1

end page

  • 35

issue

  • 5

volume

  • 26

International Standard Serial Number (ISSN)

  • 1070-5325

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(n3r) algorithm for computing the (unique) solution.

keywords

  • formal matrix product; matrix pencils; periodic qr/qz algorithm; periodic schur decomposition; sylvester and sylvester equations; systems of linear matrix equations