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A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP.

In this paper we consider the computation of the modified moments for the system of Laguerre polynomials on the real semiaxis with the Hermite weight. These moments can be used for the computation of integrals with the Hermite weight for the real semiaxis, either via product rules or via the construction of the n-point Gaussian quadrature rule with respect to this nonstandard weight using the Chebyshev algorithm. We propose a new computational method based on the construction of the null-space of a rectangular matrix derived from the three--term recurrence relation of the system of orthonormal Laguerre polynomials. It is shown that the proposed algorithm computes the modified moments with high accuracy and linear complexity. Numerical examples illustrate the effectiveness of the proposed method.

Funding Information: The authors thank the anonymous reviewers for useful comments. G.B. and V.N. are supported by an Academy of Finland grant (Suomen Akatemian päätös 331240). G.B. thanks the Alfred Kordelinin säätiö for the financial support under Grant No. 210122. P.V.D. is supported by an Aalto Science Institute Visitor Programme. Publisher Copyright: © 2022 American Physical Society. We analyze the recovery of different roles in a network modeled by a directed graph, based on the so-called Neighborhood Pattern Similarity approach. Our analysis uses results from random matrix theory to show that, when assuming that the graph is generated as a particular stochastic block model with Bernoulli probability distributions for the different blocks, then the recovery is asymptotically correct when the graph has a sufficiently large dimension. Under these assumptions there is a sufficient gap between the dominant and dominated eigenvalues of the similarity matrix, which guarantees the asymptotic correct identification of the number of different roles. We also comment on the connections with the literature on stochastic block models, including the case of probabilities of order log(n)/n where n is the graph size. We provide numerical experiments to assess the effectiveness of the method when applied to practical networks of finite size.

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 {\em 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., 29 pages

International audience; This paper addresses the problem of stability for general two-dimensional (2D) discrete-time and continuous-discrete time Lyapunov systems, where the linear matrix inequalities (LMI's) approach is applied to derive a new sufficient condition for the asymptotic stability.

Funding Information: Supported by an Academy of Finland grant (Suomen Akatemian päätös 331240).Supported by an Aalto Science Institute Visitor Programme. Publisher Copyright: © 2022 The Author(s) The notion of root polynomials of a polynomial matrix P(λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R(λ), possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R(λ), without any need to assume that R(λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R(λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ0. If λ0 is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recoverthe root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows us to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. Moreover, such definition includes, as particular cases, other definitions that have been used previously in the literature. In this way, this new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2019 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation., 49 pages

In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

We study different representations of a given rational transfer function that represents a passive (or positive real) discrete-time system. When the system is subject to perturbations, passivity or stability may be lost. To make the system robust, we use the freedom in the representation to characterize and construct optimally robust representations in the sense that the distance to non-passivity is maximized with respect to an appropriate matrix norm. We link this construction to the solution set of certain linear matrix inequalities defining passivity of the transfer function. We present an algorithm to compute a nearly optimal representation using an eigenvalue optimization technique. We also briefly consider the problem of finding the nearest passive system to a given non-passive one.

Publisher Copyright: © 2022 Informa UK Limited, trading as Taylor & Francis Group. We introduce a new family of linearizations of rational matrices, which we call block full rank linearizations. The theory of block full rank linearizations is useful as it establishes very simple criteria to determine if a pencil is a linearization of a rational matrix in a target set or in the whole underlying field, by using rank conditions. Block full rank linearizations allow us to recover locally information about zeros and poles. To recover the pole-zero information at infinity, we will define the grade of the new block full rank linearizations as linearizations at infinity and the notion of degree of a rational matrix will be used. Moreover, the eigenvectors of a rational matrix associated with its eigenvalues in a target set can be obtained from the eigenvectors of its block full rank linearizations in that set. This new family of linearizations generalizes and includes the structures appearing in most of the linearizations for rational matrices constructed in the literature. As example, we study thestructure and properties of the linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, 2021].

This paper studies generic and perturbation properties inside the linear space of $m\times (m+n)$ polynomial matrices whose rows have degrees bounded by a given list $d_1, \ldots, d_m$ of natural numbers, which in the particular case $d_1 = \cdots = d_m = d$ is just the set of $m\times (m+n)$ polynomial matrices with degree at most $d$. Thus, the results in this paper extend to a much more general setting the results recently obtained in [Van Dooren & Dopico, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.011] only for polynomial matrices with degree at most $d$. Surprisingly, most of the properties proved in [Van Dooren & Dopico, Linear Algebra Appl. (2017)], as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to $d_1, \ldots , d_m$, and with right minimal indices differing at most by one and having a sum equal to $\sum_{i=1}^{m} d_i$, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly., arXiv admin note: text overlap with arXiv:1612.03793

We revisit the notion of root polynomials, thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020] for general polynomial matrices, and show how they can efficiently be computed in the case of matrix pencils. The staircase algorithm implicitly computes so-called zero directions, as defined in [P. Van Dooren, Computation of zero directions of transfer functions, Proceedings IEEE 32nd CDC, 3132--3137, 1993]. However, zero directions generally do not provide the correct information on partial multiplicities and minimal indices. These indices are instead provided by two special cases of zero directions, namely, root polynomials and vectors of a minimal basis of the pencil. We show how to extract, starting from the block triangular pencil that the staircase algorithm computes, both a minimal basis and a maximal set of root polynomials in an efficient manner. Moreover, we argue that the accuracy of the computation of the root polynomials can be improved by making use of iterative refinement.

In this paper we analyze an indirect approach, called the Neighborhood Pattern Similarity approach, to solve the so-called role extraction problem of a large-scale graph. The method is based on the preliminary construction of a node similarity matrix which allows in a second stage to group together, with an appropriate clustering technique, the nodes that are assigned to have the same role. The analysis builds on the notion of ideal graphs where all nodes with the same role, are also structurally equivalent.