Strongly Minimal Self-Conjugate Linearizations for Polynomial and Rational Matrices

Abstract

We prove that we can always construct strongly minimal linearizations of an arbitrary rational matrix from its Laurent expansion around the point at infinity, which happens to be the case for polynomial matrices expressed in the monomial basis. If the rational matrix has a particular self-conjugate structure we show how to construct strongly minimal linearizations that preserve it. The structures that are considered are the Hermitian and skew-Hermitian rational matrices with respect to the real line, and the para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary axis. We pay special attention to the construction of strongly minimal linearizations for the particular case of structured polynomial matrices. The proposed constructions lead to efficient numerical algorithms for constructing strongly minimal linearizations. The fact that they are valid for any rational matrix is an improvement on any other previous approach for constructing other classes of structure preserving linearizations, which are not valid for any structured rational or polynomial matrix. The use of the recent concept of strongly minimal linearization is the key for getting such generality.Strongly minimal linearizations are Rosenbrock's polynomial system matrices of the given rational matrix, but with a quadruple of linear polynomial matrices (i.e. pencils) :where A(λ) is regular, and the pencils A(λ) −B(λ) and A(λ) C(λ) have no finite or infinite eigenvalues. Strongly minimal linearizations contain the complete information about the zeros, poles and minimal indices of the rational matrix and allow to recover very easily its eigenvectors and minimal bases. Thus, they can be combined with algorithms for the generalized eigenvalue problem for computing the complete spectral information of the rational matrix.

Linearizations of matrix polynomials viewed as Rosenbrock's system matrices

“…Thus, by Theorem 1.2, L(λ) contains the information about finite poles and zeros of R(λ), and is a linearization of R(λ) in the sense of [4,Definition 3.2]. More information about different definitions of linearizations of rational matrices and how to construct linear polynomial system matrices that also preserve the pole and zero information at infinity, i.e., the pole and zero information at 0 of R(1/λ), can be found, for instance, in [4,17,18,19].…”

“…In that case we say that rev 1 L(λ) and diag(rev ℓ P (λ), I s ) are equivalent at 0. In this paper the terms linearization and strong linearization of a matrix polynomial always refer to the Gohberg, Lancaster and Rodman's definitions in (2) and (3), though other non-equivalent definitions of linearizations are available in the literature [18,19]. In the last two decades, definitions (2) and (3) have been very influential in many families of linearizations that have been developed with the goal of solving unstructured and structured PEPs (see, for instance, [3,5,7,10,11,15,25,26,27,28,30,36] among many other references on this topic).…”

Revisiting the matrix polynomial greatest common divisor

“…One of the main applications of GCRD and GCLD extraction (for F = C or R) is the reduction of polynomial system quadruples {A(λ), B(λ), C(λ), D(λ)}, introduced by Rosenbrock [22] in linear system theory, and recently become of interest also in the context of nonlinear eigenvalue problems, see e.g. [6,8,19] and the references therein. Such system quadruples are a minimal representation of the (rational) transfer function R(λ) := C(λ)A(λ) −1 B(λ) + D(λ) provided the pairs {−A(λ), C(λ)} and {−A(λ), B(λ)} are right and left coprime, respectively.…”