Linearizations of matrix polynomials viewed as Rosenbrock's system matrices

Dopico, Froilán M.;Marcaida, Silvia;Quintana, María C.;Van Dooren, Paul
(2024) Linear Algebra and Its Applications — Vol. 693, p. 116-139 (2024)

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Authors
  • Dopico, Froilán M.orcid-logoUniversidad Carlos III de Madrid, Spain
    Author
  • Marcaida, SilviaUniversidad del País Vasco UPV/EHU, Spain
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  • Quintana, María C.Aalto University, Aalto, Finland
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  • Van Dooren, PaulUCLouvain
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Abstract
A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP.
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Dopico, F. M., Marcaida, S., Quintana, M. C., & Van Dooren, P. (2024). Linearizations of matrix polynomials viewed as Rosenbrock’s system matrices. Linear Algebra and Its Applications, 693, 116-139. https://doi.org/10.1016/j.laa.2023.05.010 (Original work published 2024)