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Let be open convex sets, and , , and , , be (maybe repetitive) points. Let be an analytic function. Let the interpolating polynomial be determined by the values of on the rectangular grid , , . Let and be matrices of the sizes and , respectively. The function of and can be defined by the formula where and surround the spectra and , respectively; is defined in the same way. An estimate of is given. [ABSTRACT FROM AUTHOR]

We investigate the solutions of the Sylvester-type matrix polynomial equation A(λ)X(λ) + Y (λ)B(λ) = C(λ), where A(λ), B(λ), and C(λ) are the polynomial matrices with elements in a ring of polynomials F[λ], F is a field, X(λ) and Y (λ) are unknown polynomial matrices. Solving such a matrix equation is reduced to the solving a system of linear equations... over a field F. In this case, the Kronecker product of matrices is applied. In terms of the ranks of matrices over a field F, which are constructed by the coefficients of the Sylvestertype matrix polynomial equation, the necessary and sufficient conditions for the existence of solutions X0(λ) and Y0(λ) of given degrees to the Sylvester-type matrix polynomial equation are established. The solutions of this matrix polynomial equation are constructed from the solutions of the linear equations system. As a consequence of the obtained results, we give the necessary and sufficient conditions for the existence of the scalar solutions X0 and Y0, whose entries are elements in a field F, to the Sylvester-type matrix polynomial equation. [ABSTRACT FROM AUTHOR]

We derive computable formulas for the structured backward errors of a complex number λ when considered as an approximate eigenvalue of rational matrix functions that carry a symmetry structure. We consider symmetric, skew-symmetric, Hermitian, skew-Hermitian, ∗ -palindromic, T-even, T-odd, ∗ -even, and ∗ -odd structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different. [ABSTRACT FROM AUTHOR]

. We consider a minimal realization of a rational matrix. We perturb all the coefficients of matrix polynomial and some coefficients from the realization part present in the realization form of rational matrix. We derive explicit computable formulae for backward error of approximate eigenvalues and eigenpairs of regular rational matrix. We also determine minimal perturbations for all the coefficients of matrix polynomial and some coefficients from the realization part for which approximate eigenvalues are exact eigenvalues of the perturbed rational matrix. [ABSTRACT FROM AUTHOR]

In [4] a model is presented of a finite-dimensional Pontryagin space with a symmetric operator without eigenvalues. In this note we show that this model is unique up to an equivalence relation. [ABSTRACT FROM AUTHOR]

Given a matrix polynomial A (λ) of degree d and the associated vector space of pencils DL (A) described in Mackey et al. [12] , we construct a parametrization for the set of left and right transformations that preserve the block structure of such pencils. They form a special class of structure-preserving transformations (SPTs). An SPT in that class maps DL (A) to DL (A ˜) , where A ˜ (λ) is a new matrix polynomial that is still of degree d and whose finite and infinite eigenvalues and their partial multiplicities are the same as those of A (λ). Unlike previous work on SPTs, we do not require the leading matrix coefficient of A (λ) to be nonsingular. We show that additional constraints on the parametrization lead to SPTs that also preserve extra structures in A (λ) such as symmetric, alternating, and T -palindromic structures. Our parametrization allows easy construction of SPTs that are low-rank modifications of the identity matrix. The latter transform A (λ) into a matrix polynomial A ˜ (λ) whose j th matrix coefficient A ˜ j is a low-rank modification of A j. We expect such SPTs to be one of the key tools for developing algorithms that reduce a matrix polynomial to Hessenberg form or tridiagonal form in a finite number of steps and without the use of a linearization. [ABSTRACT FROM AUTHOR]

Some time in the early 2000's, I extended the routine CompanionMatrix in the LinearAlgebra package to compute what are called linearizations of matrix polynomials. These are just univariate polynomials with matrix coefficients; isomorphically, these are matrices with univariate polynomial entries. Linearizations can be used to solve multivariate systems of equations by a number of techniques, which are "well-known" in the sense that they are in books and papers. However "well-known" they are, they deserve to be better-known, and this expository paper gives examples of some of the methods that can be used. Think of this as an extended help page for the code (which, if I am honest, is long overdue for an upgrade). [ABSTRACT FROM AUTHOR]

This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include T$$ T $$‐symmetric, T$$ T $$‐skew‐symmetric, Hermitian, skew Hermitian, T$$ T $$‐even, T$$ T $$‐odd, H$$ H $$‐even, H$$ H $$‐odd, T$$ T $$‐palindromic, T$$ T $$‐anti‐palindromic, H$$ H $$‐palindromic, and H$$ H $$‐anti‐palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real‐life applications. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for Mathematica is proposed for the computation of the asymptotic Fisher information matrix for a multivariate time series, more precisely for a controlled vector autoregressive moving average stationary process, or VARMAX process. Meanwhile, we present briefly several algorithms published in the literature and discuss the sufficient condition of invertibility of that matrix based on the eigenvalues of the process operators. The results are illustrated by numerical computations. [ABSTRACT FROM AUTHOR]

The paper presents methods for the eigenvalue localisation of regular matrix polynomials, in particular, the stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Szász inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several seconds- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable. [ABSTRACT FROM AUTHOR]

We give several new upper bounds for the eigenvalues of monic matrix polynomials with commuting coefficients by applying several numerical radius inequalities to the Frobenius companion matrices of these polynomials. [ABSTRACT FROM AUTHOR]

We extend a result proved by Dirr and Wimmer [IEEE Trans. Automat. Control 52(2007)] for polynomials to the matrix valued analytic functions and thereby obtain generalizations of some well-known results concerning the zero free regions of a class of analytic functions. [ABSTRACT FROM AUTHOR]

We are committed to the study of neutral type differential equation with delay and pairwise permutable matrices on Yang’s fractal sets R m κ (0 < κ ≤ 1 , m ∈ N) via local fractional-order calculus theory. Firstly, the fundamental solution of the matrix equation with initial condition has been presented by constructing the piecewise defined delayed matrix polynomial function on Yang’s fractal sets. Secondly, assuming the linear parts to be given by pairwise permutable constant matrices, we got the exact solution of the homogeneous initial value problem and the non-homogeneous neutral differential equation with a given initial condition. Finally, the solution of a neutral differential equation with pure delay was given by the sum of solution of homogeneous problem and a particular solution of non-homogeneous problem. The present formulation can lay a foundation for the study of system stability, controllability and oscillatory. [ABSTRACT FROM AUTHOR]

Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two: (a) provide a characterization for the inclusion relation between the closures (in the standard topology) of bundles; and (b) are the bundles open in their closure? The main goal of this paper is providing an explicit answer to these two questions. In order to get this answer, we also review and/or formalize some notions and results already existing in the literature. We also prove that bundles of matrices under similarity, as well as bundles of matrix polynomials (defined as the set of m × n matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to show that the numerical range of a linear pencil λ A + B is equal to C if and only if 0 belongs to the convex hull of the joint numerical range of A and B. We also prove that if the numerical range of a linear pencil λ A + B is equal to C and A + A ⁎ , B + B ⁎ ⩾ 0 , then A and B have a common isotropic vector. Moreover, we improve the classical result which describes Hermitian linear pencils. [ABSTRACT FROM AUTHOR]

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix G (λ) when G (λ) has the same structure. We also describe recovery of eigenvectors, minimal bases and minimal indices of G (λ) from those of the linearizations of G (λ) and show that the recovery is operation-free. [ABSTRACT FROM AUTHOR]

In this paper, basing on the theory of matricial Hamburger moment problems, we establish the intrinsic connections between the quasi-stability of a monic or comonic matrix polynomial and the Stieltjes property of a rational matrix-valued function built from the even–odd split of the original matrix polynomial. As applications of these connections, we obtain some new criteria for quasi-stable matrix polynomials and Hurwitz stable matrix polynomials, respectively. [ABSTRACT FROM AUTHOR]

We consider systems of second-order, quasilinear, ordinary differential equations with an identically degenerate matrix coefficient of the principal term and with well-posed initial conditions. Fundamental differences between such problems and systems of ordinary differential equations solved with respect to the second derivative are indicated. In terms of matrix polynomials, we formulate conditions of the existence and uniqueness of solutions of such problems in a neighborhood of the starting point. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for Mathematica is proposed for the computation of the asymptotic Fisher information matrix for a multivariate time series, more precisely for a controlled vector autoregressive moving average stationary process, or VARMAX process. Meanwhile, we present briefly several algorithms published in the literature and discuss the sufficient condition of invertibility of that matrix based on the eigenvalues of the process operators. The results are illustrated by numerical computations.

A multiscaling equation in the Fourier domain accommodates a trigonometric matrix polynomial. This trigonometric matrix polynomial is known as the symbol function. The existence and properties of a multiscaling function, which is the solution of a multiscaling equation, depend on the symbol function. It is possible to construct symbol functions corresponding to compactly supported and symmetric multiscaling functions from standard pairs. A standard pair carries the spectral information about the symbol function. In this paper, we briefly explain the construction of compactly supported and symmetric multiscaling functions and the corresponding mulitwavelets using standard pairs. We derive the necessary as well as sufficient condition, on the eigenspace of the square matrix in the standard pair, for the existence of a symbol function corresponding to a multiscaling equation with a compactly supported solution. We create a pseudo bi-orthogonal pair of symmetric and compactly supported multiscaling functions and the corresponding multiwavelets using standard pairs. [ABSTRACT FROM AUTHOR]

Consider a higher order state space system and associated system matrix S(λ). The aim of this paper is to linearize the system preserving system characteristics. That is, linearization preserving system characteristics (e.g, controllability, observability, various zeros and transfer function) for analysis of higher order state-space systems. In particular, we introduce Fiedler-like linearizations (Fiedler linearizations, proper generalized Fiedler (PGF) linearizations) of the system matrix S(λ) to study zeros of higher order system. Further, we show that the linearized systems are strict system equivalent to the higher order systems and hence preserve system characteristics of the original systems. We show that the PGF pencils of S(λ) provide a class of structure-preserving linearizations of S(λ). We study recovery of zero directions of higher order state space system from those of the linearizations. That is, the zero directions of the transfer functions associated to higher order state space system are recovered from the eigenvectors of the Fiedler pencils without performing any arithmetic operations. [ABSTRACT FROM AUTHOR]

In this paper, we prove a more general result concerning the location of the eigenvalues of a matrix polynomial in an annulus from which we deduce an interesting result due to Higham and Tisseur [11]. Several other known results have been extended to matrix polynomials, which in particular include extension and generalization of a classical result of Cauchy [4]. We also present two examples of matrix polynomials to show that the bounds obtained are close to the actual bounds. [ABSTRACT FROM AUTHOR]

In this paper, we focus on discussing diagonal solutions and general solutions of second-order matrix polynomial equation of high degree in complex field. By characterizing some algebraic properties of the mentioned two types of the solutions, we present sufficient conditions that a general second-order matrix polynomial equation has diagonal solutions or general solutions. Analytic expressions of the solutions, as well as the corresponding algorithms for finding the solutions are provided. An example is given so as to verify the theoretical results we have derived. [ABSTRACT FROM AUTHOR]

This paper focuses on stability analysis and stabilization of a switched system under average dwell time criteria in the continuous-time domain. The matrix polynomial approach is applied to the switched system to construct a continuous Lyapunov function and design a more effective controller, which can not only improve the system performance but also lessen the system conservativeness. The stability problem is studied with square matricial representation for the first time. The new method is more applicable than previous work under average dwell time switching with the time-varying controller gains. In addition, new sufficient conditions of stability and stabilization are derived to guarantee the global uniform exponential stability of the switched system. A numerical example is provided to show the advantages of the new method.

A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations. [ABSTRACT FROM AUTHOR]

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived, and the eigenvalues are localised. Four instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product HX of a fixed diagonal matrix H and the Wigner matrix X and two special matrix polynomials of higher degree. The results are illustrated with various examples and numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper we show that stationary and non-stationary multivariate continuous-time ARMA (MCARMA) processes have the representation as a sum of multivariate complex-valued Ornstein-Uhlenbeck processes under some mild assumptions. The proof benefits from properties of rational matrix polynomials. A conclusion is an alternative description of the autocovariance function of a stationary MCARMA process. Moreover, that representation is used to show that the discrete-time sampled MCARMA(p, q) process is a weak VARMA(p, p - 1) process if second moments exist. That result complements the weak VARMA(p, p - 1) representation derived in Chambers and Thornton (2012). In particular, it relates the right solvents of the autoregressive polynomial of the MCARMA process to the right solvents of the autoregressive polynomial of the VARMA process; in the one-dimensional case the right solvents are the zeros of the autoregressive polynomial. Finally, a factorization of the sample autocovariance function of the noise sequence is presented which is useful for statistical inference. [ABSTRACT FROM AUTHOR]

In this paper, the notions of pseudofield of values and joint pseudofield of values of matrix polynomials are introduced and some of their algebraic and geometrical properties are studied. Moreover, the relationship between the pseudofield of values of a matrix polynomial and the pseudofield of values of its companion linearization is stated, and then some properties of the augmented field of values of basic A-factor block circulant matrices are investigated.

In this paper, new conditions of the stability, stabilization and L2-gain performance of periodic piecewise systems are proposed. Both the continuous and discontinuous Lyapunov functions with dwell-time related time-varying Lyapunov matrix polynomial are adopted, and methods guaranteeing the positive and negative definiteness of a matrix polynomial are introduced. Exponential stability conditions are derived based on continuous and discontinuous Lyapunov functions, respectively. A stabilizing controller with time-varying controller gain is designed with continuous Lyapunov function and the weighted L2-gain performance based on the discontinuous Lyapunov matrix polynomial is studied as well. Numerical examples are used to verify the effectiveness of the proposed methods.

In this paper, basing on the theory of matricial Hamburger moment problems, we establish the intrinsic connections between the quasi-stability of a monic or comonic matrix polynomial and the Stieltjes property of a rational matrix-valued function built from the even–odd split of the original matrix polynomial. As applications of these connections, we obtain some new criteria for quasi-stable matrix polynomials and Hurwitz stable matrix polynomials, respectively.

This paper concentrates on the problem of continuous-time switched linear systems with all subsystems unstable under average dwell time (ADT) criteria. Inspired by the matrix polynomial approach, a new method is proposed to further lessen conservativeness and improve system performance. The new method is based on a matrix polynomial and the discretized Lyapunov function (DLF) technique. Using the matrix polynomial, the DLF is successfully established and applied to switched systems. Based on this function, convex sufficient conditions are derived, thereby ensuring the global uniform exponential stability of the system. In addition, the method can be expanded to uncertain systems. Finally, a numerical example is presented to illustrate the potential of the proposed approach. • The method is first used in switched systems with the discretized Lyapunov function. • The sufficient convex condition is derived with the matrix polynomial method. • The method can further lessen conservativeness compared with existing research. • The method can be expanded to uncertain systems. [ABSTRACT FROM AUTHOR]

The classical Eneström-Kakeya theorem establishes explicit upper and lower bounds on the zeros of a polynomial with positive coefficients and has been generalized for positive definite matrix polynomials by several authors. Recently, extensions that improve the (scalar) Eneström-Kakeya theorem were obtained with a transparent and unified approach using just a few tools. Here, the same tools are used to generalize these extensions to positive definite matrix polynomials, while at the same time generalizing the tools themselves. In the process, a framework is developed that can naturally generate additional similar results.

Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications. Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.

In this article, a novel nonfragile controller design scheme is developed for a class of periodic piecewise systems with linear time-varying subsystems. Two types of norm-bounded controller perturbations, including additive and multiplicative ones, are considered and partially characterized by periodic piecewise time-varying parameters. Using a new matrix polynomial lemma, the problems of nonfragile controller synthesis for periodic piecewise time-varying systems (PPTVSs) are made amenable to convex optimization based on the favorable property of a class of matrix polynomials. Depending on selectable divisions of subintervals, sufficient conditions of the stability and nonfragile controller design are proposed for PPTVSs. Case studies based on a multi-input multi-output PPTVS and a mass-spring-damper system show that the proposed control schemes can effectively guarantee the close-loop stability and accelerate the convergence under controller perturbations, with more flexible periodic time-varying controller gains than those obtained by the existing methods. [ABSTRACT FROM AUTHOR]

We introduce the concept of a canonical subspace of C d z and among other results prove the following statements. An operator in a finite-dimensional vector space has no eigenvalues if and only if it is similar to the operator of multiplication by the independent variable on a canonical subspace of C d z. An operator in a finite-dimensional Pontryagin space is symmetric and has no eigenvalues if and only if it is isomorphic to the operator of multiplication by the independent variable in a canonical subspace of C d z with an inner product determined by a full matrix polynomial Nevanlinna kernel. [ABSTRACT FROM AUTHOR]

We derive a new family of polynomial multipliers that improve the Pellet radii of matrix polynomials, which determine an annulus in the complex plane devoid of eigenvalues, and that generalize and improve previously obtained results, while relaxing the conditions under which these hold. As a special limit case we also obtain an improvement of the Cauchy radius, which is an upper bound on the moduli of the eigenvalues. Results for the special case of scalar polynomials are presented. [ABSTRACT FROM AUTHOR]

The problem of finding the distance from a given n×n matrix polynomial of degree k to the set of matrix polynomials having the elementary divisor (λ-λ0)j,j⩾r, for a fixed scalar λ0 and 2⩽r⩽kn is considered. It is established that polynomials that are not regular are arbitrarily close to a regular matrix polynomial with the desired elementary divisor. For regular matrix polynomials the problem is shown to be equivalent to finding minimal structure preserving perturbations such that a certain block Toeplitz matrix becomes suitably rank deficient. This is then used to characterize the distance via two different optimizations. The first one shows that if λ0 is not already an eigenvalue of the matrix polynomial, then the problem is equivalent to computing a generalized notion of a structured singular value. The distance is computed via algorithms like BFGS and Matlab's globalsearch algorithm from the second optimization. Upper and lower bounds of the distance are also derived and numerical experiments are performed to compare them with the computed values of the distance. [ABSTRACT FROM AUTHOR]

We prove that the determinant of a matrix polynomial with Hermitian positive definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients. [ABSTRACT FROM AUTHOR]

The standard way to solve polynomial eigenvalue problems is through linearizations. The family of Fiedler linearizations, which includes the classical Frobenius companion forms, presents many interesting properties from both the theoretical and the applied point of view. These properties make the Fiedler pencils a very attractive family of linearizations to be used in the solution of polynomial eigenvalue problems. However, their numerical features for general matrix polynomials had not yet been fully investigated. In this paper, we analyze the backward error of eigenpairs and the condition number of eigenvalues of Fiedler linearizations in the solution of polynomial eigenvalue problems. We get bounds for: (a) the ratio between the backward error of an eigenpair of the matrix polynomial and the backward error of the corresponding (computed) eigenpair of the linearization, and (b) the ratio between the condition number of an eigenvalue in the linearization and the condition number of the same eigenvalue in the matrix polynomial. A key quantity in these bounds is \rho , the ratio between the maximum norm of the coefficients of the polynomial and the minimum norm of the leading and trailing coefficient. If the matrix polynomial is well scaled (i. e., all its coefficients have a similar norm, which implies ρ ≈ 1), then solving the Polynomial Eigenvalue Problem with any Fiedler linearization will give a good performance from the point of view of backward error and conditioning. In the more general case of badly scaled matrix polynomials, dividing the coefficients of the polynomial by the maximum norm of its coefficients allows us to get better bounds. In particular, after this scaling, the ratio between the eigenvalue condition number in any two Fiedler linearizations is bounded by a quantity that depends only on the size and the degree of the polynomial. We also analyze the effect of parameter scaling in these linearizations, which improves significantly the backward error and conditioning in some cases where ρ is large. Several numerical experiments are provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

olutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach has a potential to be more suitable for applications. Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given. [ABSTRACT FROM AUTHOR]

We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of polynomial eigenvalue problems (PEPs). By using the homogeneous formulation of PEPs, we are able to obtain two clear and simple results. First, we show that if the matrix inducing the Möbius transformation is well-conditioned, then such transformation approximately preserves the eigenvalue condition numbers and backward errors when they are defined with respect to perturbations of the matrix polynomial which are small relative to the norm of the whole polynomial. However, if the perturbations in each coefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backward errors are preserved approximately by the Möbius transformations induced by well-conditioned matrices only if a penalty factor, depending on the norms of those matrix coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation of the PEP is used. [ABSTRACT FROM AUTHOR]

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed. [ABSTRACT FROM AUTHOR]

Walks around a graph are studied in a wide range of fields, from graph theory and stochastic analysis to theoretical computer science and physics. In many cases it is of interest to focus on non-backtracking walks; those that do not immediately revisit their previous location. In the network science context, imposing a non-backtracking constraint on traditional walk-based node centrality measures is known to offer tangible benefits. Here, we use the Hashimoto matrix construction to characterize, generalize and study such non-backtracking centrality measures. We then devise a recursive extension that systematically removes triangles, squares and, generally, all cycles up to a given length. By characterizing the spectral radius of appropriate matrix power series, we explore how the universality results on the limiting behaviour of classical walk-based centrality measures extend to these non-cycling cases. We also demonstrate that the new recursive construction gives rise to practical centrality measures that can be applied to largescale networks. [ABSTRACT FROM AUTHOR]

In this paper we have factorized matrix polynomials into a complete set of spectral factors using a new design algorithm, and with some systematic procedures a complete set of block roots (solvents) have been obtained. The newly developed procedure is just an extension of the (scalar) Horner method to its block form for use in the computation of the block roots of matrix polynomial, the block-Horner method bringing a local iterative nature, faster conver- gence, nested programmable scheme, needless of any prior knowledge of the matrix polynomial, with the only one inconvenience, which is the strong dependence on the initial guess. In order to avoid this trap, we proposed a combination of two computational procedures, for which the complete program starts with the right block-Q.D. algorithm. It is then followed by a refinement of the right factor by block-Horner's algorithm. This results in the global nature of the program, which is faster in execution, has well defined initial conditions, and good convergence in much less time. [ABSTRACT FROM AUTHOR]

This paper is a continuation of a recent investigation by Zhan and Dyachenko (2021) on the Hurwitz stability of monic matrix polynomials with algebraic techniques. By improving an inertia formula for matrix polynomials with respect to the imaginary axis, we show that, under some conditions, the quasi-stability of a monic matrix polynomial can be tested via the Hermitian nonnegative definiteness of two block Hankel matrices built from its matricial Markov parameters. Moreover, for the so-called doubly monic matrix polynomials, the quasi-stability criteria can be formulated in a much simpler form. In particular, the relationship between Hurwitz stable matrix polynomials and Stieltjes positive definite matrix sequences established in Zhan and Dyachenko (2021) is included as a special case. [ABSTRACT FROM AUTHOR]