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The Cauchy radius of a scalar polynomial is an upper bound on the magnitude of its zeros, and it is optimal among all bounds depending only on the moduli of the coefficients. It has the disadvantage of being implicit because it requires the solution of a nonlinear equation. In this note, simple and explicit upper bounds are derived that are useful approximations to the Cauchy radius. The general approach to obtain the aforementioned bounds is to embed scalar polynomials into the larger framework of their generalization to matrix polynomials and then use bounds on the eigenvalues of the latter. [ABSTRACT FROM AUTHOR]

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]

This paper concerns the study of matrix polynomials of arbitrary degree. In terms of L (λ) = λ r I - ∑ j = 1 r λ r - j C j with or without commuting coefficients ( C i C j = C j C i , o r C i C j ≠ C j C i f o r C i ∈ C t × t , i , j = 1 , ... , r ) by determinant of block Toeplitz-Hessenberg matrices. [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]

Let σ0,σ1,⋯,σn be a set of n+ 1 distinct real numbers (i.e., σi≠σj, for i≠j) and F0,F1,⋯ ,Fn, be given real s × r matrices, we know that there exists a unique s × r matrix polynomial Pn(λ) of degree n such that Pn(σi) = Fi, for i = 0,1,⋯ ,n, Pn is the matrix interpolation polynomial for the set {(σi,Fi),i = 0,1,⋯ ,n}. The matrix polynomial Pn(λ) can be computed by using the Lagrange formula or the barycentric method. This paper presents a new method for computing matrix interpolation polynomials. We will reformulate the Lagrange matrix interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Recursive Matrix Polynomial Interpolation Algorithm (RMPIA) in full and simplified versions, and some properties of this algorithm will be studied. Cost and storage of this algorithm with the classical formulas will be studied and some examples will also be given. [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]

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, 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 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]

In this paper, we consider the linearly structured partial polynomial inverse eigenvalue problem (LPPIEP) of constructing the matrices A i ∈ R n × n for i = 0 , 1 , 2 , ... , (k − 1) of specified linear structure such that the matrix polynomial P (λ) = λ k I n + ∑ i = 0 k − 1 λ i A i has the m (1 ⩽ m ⩽ k n) prescribed eigenpairs as its eigenvalues and eigenvectors. Many practical applications give rise to linearly structured matrix polynomials. Typical linearly structured matrices are symmetric, skew-symmetric, tridiagonal, diagonal, pentagonal, Hankel, Toeplitz, etc. Therefore, construction of the matrix polynomial with the aforementioned structures is an important but challenging aspect of the polynomial inverse eigenvalue problem (PIEP). In this paper, a necessary and sufficient condition for the existence of solution to this problem is derived. Additionally, we characterize the class of all solutions to this problem by giving the explicit expressions of the solutions. It should be emphasized that the results presented in this paper resolve some important open problems in the area of PIEP namely, the inverse eigenvalue problems for structured matrix polynomials such as symmetric, skew-symmetric, alternating matrix polynomials as pointed out by De Terán et al. (2015). Further, we study sensitivity of solution to the perturbation of the eigendata. An attractive feature of our solution approach is that it does not impose any restriction on the number of eigendata for computing the solution of LPPIEP. Towards the end, the proposed method is validated with various numerical examples on a spring mass problem. [ABSTRACT FROM AUTHOR]