TY - JOUR

T1 - Structured strong l-ifications for structured matrix polynomials in the monomial basis

AU - De Terán, Fernando

AU - Hernando, Carla

AU - Pérez, Javier

N1 - Publisher Copyright:
© 2021, International Linear Algebra Society. All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - (Anti-)Palindromic

KW - (Skew)-Symmetric

KW - (Skew-)Hermitian

KW - Alternating

KW - Companion forms

KW - Eigenvalues

KW - Eigenvectors

KW - L-ifications

KW - Linearizations

KW - Matrix pencils

KW - Matrix polynomials

KW - Polynomial eigenvalue problems

KW - Structured matrix polynomials

UR - http://www.scopus.com/inward/record.url?scp=85101944873&partnerID=8YFLogxK

U2 - 10.13001/ela.2021.5473

DO - 10.13001/ela.2021.5473

M3 - Article

AN - SCOPUS:85101944873

SN - 1537-9582

VL - 37

SP - 35

EP - 71

JO - Electronic Journal of Linear Algebra

JF - Electronic Journal of Linear Algebra

ER -