On vector spaces of linearizations for matrix polynomials in orthogonal bases

TL;DR

This paper analyzes vector spaces of potential linearizations for matrix polynomials in orthogonal bases, providing concise characterizations, criteria, and structural insights into their properties and equivalences.

Contribution

It offers new algebraic characterizations and criteria for linearizations in vector spaces, extending previous results to orthogonal bases and exploring structural similarities.

Findings

Characterization of all pencils in M1(P)

Criteria for strong linearizations in M1(P)

Structural similarities between classical and generalized pencils

Abstract

Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential linearizations for P are analyzed. All pencils in M1(P) are characterized concisely. Moreover, several easy to check criteria whether a pencil in M1(P) is a (strong) linearization of P are given. The equivalence of some of them to the Z-rank-condition (see Mackey et al. 2006) is pointed out. Results on the vector space dimensions, the genericity of linearizations in and the form of block-symmetric pencils are derived in a new way on a basic algebraic level. Throughout the paper, structural resemblances between the matrix pencils in L1 , i.e. the results obtained in Mackey et al. 2006, and their generalized versions are pointed out.