Structured strong l-ifications for structured matrix polynomials in the monomial basis

Fernando De Terán, Carla Hernando, Javier Pérez

Research output: Contribution to journalArticlepeer-review

Abstract

In the framework of Polynomial Eigenvalue Problems (PEPs), most of the matrix polynomials arising in applications are structured polynomials (namely, (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve PEPs is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix polynomials of a fixed degree, known as companion linearizations. It is well known, however, that it is not possible to construct companion linearizations that preserve any of the previous structures for matrix polynomials of even degree. This motivates the search for more general companion forms, in particular companion l-ifications. In this paper, we present, for the first time, a family of (generalized) companion l-ifications that preserve any of these structures, for matrix polynomials of degree k = (2d+1)l. We also show how to construct sparse l-ifications within this family. Finally, we prove that there are no structured companion quadratifications for quartic matrix polynomials.

Original languageEnglish
Pages (from-to)35-71
Number of pages37
JournalElectronic Journal of Linear Algebra
Volume37
DOIs
StatePublished - 2021

Keywords

  • (Anti-)Palindromic
  • (Skew)-Symmetric
  • (Skew-)Hermitian
  • Alternating
  • Companion forms
  • Eigenvalues
  • Eigenvectors
  • L-ifications
  • Linearizations
  • Matrix pencils
  • Matrix polynomials
  • Polynomial eigenvalue problems
  • Structured matrix polynomials

Fingerprint

Dive into the research topics of 'Structured strong l-ifications for structured matrix polynomials in the monomial basis'. Together they form a unique fingerprint.

Cite this