Robustness and perturbations of minimal bases

Abstract

Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important role in control theory, linear systems theory, and coding theory. It is a common practice to arrange the vectors of any minimal basis as the rows of a polynomial matrix and to call such matrix simply a minimal basis. Very recently, minimal bases, as well as the closely related pairs of dual minimal bases, have been applied to a number of problems that include the solution of general inverse eigenstructure problems for polynomial matrices, the development of new classes of linearizations and ℓ-ifications of polynomial matrices, and backward error analyses of complete polynomial eigenstructure problems solved via a wide class of strong linearizations. These new applications have revealed that although the algebraic properties of minimal bases are rather well understood, their robustness and the behavior of the corresponding dual minimal bases under perturbations have not yet been explored in the literature, as far as we know. Therefore, the main purpose of this paper is to study in detail when a minimal basis M (λ) is robust under perturbations, i.e., when all the polynomial matrices in a neighborhood of M (λ) are minimal bases, and, in this case, how perturbations of M (λ) change its dual minimal bases. In order to study such problems, a new characterization of whether or not a polynomial matrix is a minimal basis in terms of a finite number of rank conditions is introduced and, based on it, we prove that polynomial matrices are generically minimal bases with very specific properties. In addition, some applications of the results of this paper are discussed.

Block minimal bases ℓ-ifications of matrix polynomials

“…This motivates Lemma 5.2, where convolution matrices † will be used. For any matrix polynomial Q(λ) = q i=0 Q i λ i of grade q and † Convolution matrices are called Sylvester matrices in [24]. More specifically, the convolution matrix C j (Q) is the Sylvester matrix S j+1 (Q), j = 0, 1, .…”

“…Since then, they have played an important role in multivariable linear systems theory, coding theory, control theory, and in the spectral theory of rational and polynomial matrices. For detailed introductions to minimal bases, their algebraic properties, computational schemes for constructing such bases from arbitrary polynomial bases, their robustness under perturbations, and their role in the singular structure of singular rational and polynomial matrices, we refer the reader to the classical works [29,34,53], the works [3,28,24] and [39], where an elegant approach to minimal bases via filtrations is presented.…”

“…We mention in passing that a potential advantage -from the numerical point of view-of the ℓ-ifications introduced in this work is that the minimal bases involved in their construction are particular instances of the so called full-Sylvester-rank minimal bases [24]. Full-Sylvester-rank minimal bases are the only kind of minimal bases for which it is possible to perform a perturbation analysis [24] when fixing the grade of the minimal basis.…”

“…We mention in passing that a potential advantage -from the numerical point of view-of the ℓ-ifications introduced in this work is that the minimal bases involved in their construction are particular instances of the so called full-Sylvester-rank minimal bases [24]. Full-Sylvester-rank minimal bases are the only kind of minimal bases for which it is possible to perform a perturbation analysis [24] when fixing the grade of the minimal basis. These perturbation results have been successfully used for performing a rigorous backward error analysis of solving CPEP's by using block Kronecker linearizations [22] or by using the ℓ-ifications introduced by De Terán, Dopico and Van Dooren [16,24].…”

“…Full-Sylvester-rank minimal bases are the only kind of minimal bases for which it is possible to perform a perturbation analysis [24] when fixing the grade of the minimal basis. These perturbation results have been successfully used for performing a rigorous backward error analysis of solving CPEP's by using block Kronecker linearizations [22] or by using the ℓ-ifications introduced by De Terán, Dopico and Van Dooren [16,24]. Thus, we expect that our framework for constructing ℓ-ifications from dual minimal bases, together with the perturbation results for full-Sylvesterrank minimal bases [24], will allow us to extend in a future work the backward error results in [24] for the ℓ-ifications in [16] to a wider family of ℓ-ifications.…”