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The standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil, known as a strong linearization. This process transforms the problem into an equivalent generalized eigenvalue problem. However, there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials. This has motivated the idea of a strong l-ification of a matrix polynomial, which is a matrix polynomial of degree at most l having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial. We present in this work a novel method for constructing strong l-ifications of matrix polynomials of size m × n and grade d when l d , and l divides nd or md. This method is based on a family called “strong block minimal bases matrix polynomials”, and relies heavily on properties of dual minimal bases. We show how strong block minimal bases l-ifications can be constructed from the coefficients of a given matrix polynomial P ( λ ) . We also show that these l-ifications satisfy many desirable properties for numerical applications: they are strong l-ifications regardless of whether P ( λ ) is regular or singular, the minimal indices of the l-ifications are related to those of P ( λ ) via constant uniform shifts, and eigenvectors and minimal bases of P ( λ ) can be recovered from those of any of the strong block minimal bases l-ifications. In the special case where l divides d, we introduce a subfamily of strong block minimal bases matrix polynomials named “block Kronecker matrix polynomials”, which is shown to be a fruitful source of companion l-ifications.

We provide an algorithm for constructing strong l-ifications of a given matrix polynomial P ( λ ) of degree d and size m × n using only the coefficients of the polynomial and the solution of linear systems of equations. A strong l-ification of P ( λ ) is a matrix polynomial of degree l having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial P ( λ ) . All explicit constructions of strong l-ifications introduced so far in the literature have been limited to the case where l divides d, though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible. Based on recent developments on dual polynomial minimal bases, we present a general construction of strong l-ifications for wider choices of the degree l, namely, when l divides one of nd or md (and d ≥ l ). In the case where l divides nd (respectively, md), the strong l-ifications we construct allow us to easily recover the minimal indices of P ( λ ) . In particular, we show that they preserve the left (resp., right) minimal indices of P ( λ ) , and the right (resp., left) minimal indices of the l-ification are the ones of P ( λ ) increased by d − l (each). Moreover, in the particular case l divides d, the new method provides a companion l-ification that resembles very much the companion l-ifications already known in the literature.

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 x n) and Z2(λ)_(m x n), respectively, then z_1and z_2are said to be dual minimal bases if the subspaces have complementary dimension, i.e., k+m=n, and Z_1 (λ) Z_2^T (λ)=0. In other words, each z_j (λ) provides a minimal basis for the nullspace of the other. It has long been known that for any dual minimal bases z_1 (λ) and z_2 (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. Supported by National Science Foundation grant DMS-1016224, and by Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542. Publicado

The concept of linearization is fundamental for theory, applications, and spectral computations related to matrix polynomials. However, recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure-preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive, because some structured polynomials do not have any linearization with the same structure. This phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations. Motivated by this fact, we introduce equivalence relations that allow the possibility of matrix polynomials (with coefficients in an arbitrary field) to be equivalent, with the same spectral structure, but have different sizes and degrees. These equivalence relations are directly modeled on the notion of linearization, and consequently inherit the simplicity, applicability, and most relevant properties of linearizations; simultaneously, though, they are much more flexible in the possible degrees of equivalent polynomials. This flexibility allows us to define in a unified way the notions of quadratification and l-ification, to introduce the concept of companion form of arbitrary degree, and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second Frobenius companion forms. The properties of l-ifications are studied in depth; in this process a fundamental result on matrix polynomials, the “Index Sum Theorem”, is recovered and extended to arbitrary fields. Although this result is known in the systems theory literature for real matrix polynomials, it has remained unnoticed by many researchers. It establishes that the sum of the (finite and infinite) partial multiplicities, together with the (left and right) minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial. The “Index Sum Theorem” turns out to be a key tool for obtaining a number of significant results: on the possible sizes and degrees of l-ifications and companion forms, on the minimal index preservation properties of companion forms of arbitrary degree, as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree. This paper presents many new results, blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields. Therefore we have written the paper in an expository and self-contained style that makes it accessible to a wide variety of readers.