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Let $\cL = (\cL_1,\cL_2)$ be a list consisting of a sublist $\cL_1$ of powers of irreducible (monic) scalar polynomials over an algebraically closed field $\FF$, and a sublist $\cL_2$ of nonnegative integers. For an arbitrary such list $\cL$, we give easily verifiable necessary and sufficient conditions for $\cL$ to be the list of elementary divisors and minimal indices of some $T$-palindromic quadratic matrix polynomial with entries in the field $\FF$. For $\cL$ satisfying these conditions, we show how to explicitly construct a $T$-palindromic quadratic matrix polynomial having $\cL$ as its structural data; that is, we provide a $T$-palindromic quadratic realization of $\cL$. Our construction of $T$-palindromic realizations is accomplished by taking a direct sum of low bandwidth $T$-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of $T$-palindromic quadratic polynomials is that all even grade $T$-palindromic matrix polynomials have a $T$-palindromic strong quadratification. Finally, using a particular M\"{o}bius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with $T$-even structure.

We introduce a new family of strong linearizations of matrix polynomials---which we call "block Kronecker pencils"---and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This stability analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. The global backward error analysis in this work presents for the first time the following key properties: it is a rigurous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of "strong block minimal bases pencils", which are robust under certain perturbations and, so, include certain perturbations of block Kronecker pencils. We hope that this robustness property will allow us to extend the results in this paper to other contexts., this article supersedes MIMS Eprint 2016.34

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.

The development of new classes of linearizations of square matrix polynomials that generalize the classical first and second Frobenius companion forms has attracted much attention in the last decade. Research in this area has two main goals: finding linearizations that retain whatever structure the original polynomial might possess, and improving properties that are essential for accurate numerical computation, such as eigenvalue condition numbers and backward errors. However, all recent progress on linearizations has been restricted to square matrix polynomials. Since rectangular polynomials arise in many applications, it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials. In this paper, the family of Fiedler linearizations is extended from square to rectangular matrix polynomials, and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials. Fiedler linearizations are one of the most important classes of linearizations introduced in recent years, but their generalization to rectangular polynomials is nontrivial, and requires a completely different approach to the one used in the square case. To the best of our knowledge, this is the first class of new linearizations that has been generalized to rectangular polynomials.

The proceeding at: Joint ALAMA-GAMM/ANLA 2014 Meeting, took place 2014, July 14-16, in Barcelona (Spain). We present necessary and su cient conditions for the existence of a matrix polynomial when its degree, its nite and in nite elementary divisors, and its left and right minimal indices are prescribed. These conditions hold for arbitrary in nite elds and are determined mainly by the \index sum theorem", which is a fundamental relationship between the rank, the degree, the sum of all partial multiplicities, and the sum of all minimal indices of any matrix polynomial. The proof developed for the existence of such polynomial is constructive and, therefore, solves a very general inverse problem for matrix polynomials with prescribed complete eigenstructure. This result allows us to x the problem of the existence of (l)-ifications of a given matrix polynomial, as well as to determine all their possible sizes and eigenstructures. This research was partially supported by the Ministerio de Economía y Competitividad of Spain through grant MTM-2012-32542 and by the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. 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.