Polynomial zigzag matrices, dual minimal bases, and the realization of completely singular polynomials

Abstract

Minimal bases of rational vector spaces are a well known and important tool in systems theory. If minimal bases for two subspaces of rational n-space are displayed as the rows of polynomial matrices Z1(λ) k×n and Z2(λ)m×n, respectively, then Z1 and Z2 are said to be dual minimal bases if the subspaces have complementary dimension, i.e., k + m = n, and Z 1 (λ)Z T 2 (λ) = 0. In other words, each Zj(λ) provides a minimal basis for the nullspace of the other. It has long been known that for any dual minimal bases Z1(λ) and Z2(λ), the row degree sums of Z1 and Z2 are the same. In this paper we show that this is the only constraint on the row degrees, thus characterizing the possible row degrees of dual minimal bases. The proof is constructive, making extensive use of a new class of sparse, structured polynomial matrices that we have baptized zigzag matrices. Another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices -given a list of left and right minimal indices and a desired degree d, does there exist a completely singular matrix polynomial (i.e., a matrix polynomial with no elementary divisors whatsoever) of degree d having exactly the prescribed minimal indices? We show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices. The constructed realization is simple, and explicitly displays the desired minimal indices in a fashion analogous to the classical Kronecker canonical form of singular pencils.

Block minimal bases ℓ-ifications of matrix polynomials

“…We recall in Definition 3.4 the concept of dual minimal bases. These bases were introduced in [28], and named "dual minimal bases" in [21]. Remark 3.…”

Robustness and perturbations of minimal bases II: The case with given row degrees

“…Next, we introduce the concept of dual minimal bases, whose origins can be found in [19,Section 6] and that has played a key role in a number of recent applications (see [29] and [10] for more information). In the language of null-spaces of polynomial matrices, observe that M (λ) is a minimal basis of N ℓ (N (λ) T ) and that N (λ) T is a minimal basis of N r (M (λ)).…”

“…The next theorem reveals a fundamental relationship between the row degrees of dual minimal bases. Its first part was proven in [19], while the second (converse) part has been proven very recently in [10]. m×(m+n) and N (λ) ∈ F[λ] n×(m+n) be dual minimal bases with row degrees (η 1 , .…”

“…Therefore, their definition, properties, and many of their practical applications can be found in classical references on these subjects, as, for instance, the ones by Wolovich [30], Kailath [23], and Forney [19], although the concept of minimal bases is much older and, as far as we know, it was introduced for the first time in the famous paper by Dedekind and Weber [5]. Recently, minimal bases, and the closely related notion of pairs of dual minimal bases, have been applied to some problems that have attracted considerable attention in the last years as, for instance, in the solution of inverse complete eigenstructure problems for polynomial matrices [10,11], in the development of new classes of linearizations and ℓ-ifications of polynomial matrices [12,14,24,26], in the explicit construction of linearizations of rational matrices [1], and in the backward error analysis of complete polynomial eigenvalue problems solved via different classes of linearizations [14,25].Some of the applications mentioned in the previous paragraph motivated the development in the recent paper [29] of robustness and perturbation results of minimal bases, which had not been explored before in the literature. The study of any perturbation problem for polynomial matrices requires as a first step to fix the set of allowable perturbations and, with this purpose, the reference [29] considers perturbations whose only constraint is that they do not increase the degree d of the m × (m + n) given minimal basis that is perturbed.…”

Robustness and perturbations of minimal bases

“…Very recently, minimal bases, and the closely related notion of pairs of dual minimal bases, have been applied to the solution of some problems that have attracted the attention of many researchers in the last fifteen years. For instance, minimal bases have been used (1) in the solution of inverse complete eigenstructure problems for polynomial matrices (see [7,8] and the references therein), (2) in the development of new classes of linearizations and ℓ-ifications of polynomial matrices [5,9,10,11,12,18,22], which has allowed to recognize that many important linearizations commonly used in the literature are constructed via dual minimal bases (including the classical Frobenius companion forms), (3) in the explicit construction of linearizations of rational matrices [1], and (4) in the backward error analysis of complete polynomial eigenvalue problems solved via the so-called "block Kronecker linearizations" of polynomial matrices [10,Section 6], which include the interesting class of Fiedler linearizations (see [5,10] for references on this class of linearizations), but do not include most of the linearizations and ℓ-ifications that can be constructed from minimal bases [5,9,10,11,12,18,22]. See also [19] for additional references on the role played by dual minimal bases in backward error analyses.…”