Polynomial Matrix Research Articles

Explicit Relation Between Two Resolvent Matrices of the Truncated Hausdorff Matrix Moment Problem

We consider the explicit relation between two resolvent matrices related to the truncated Hausdorff matrix moment problem (THMM) in the case of an even and odd number of moments. This relation is described with the help of four families of orthogonal matrix polynomials on the finite interval [a, b] and their associated second kind polynomials.

New bounds for the Eigenvalues of Matrix Polynomials

We employ several numerical radius inequalities to the square of the Frobenius companion matrices of monic matrix polynomials to provide new bounds for the eigenvalues of these polynomials.

On the moments of moments of random matrices and Ehrhart polynomials

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size N goes to infinity. These quantities depend on two parameters k and q and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size N. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey [7]. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.

On the Rellich eigendecomposition of para-Hermitian matrices and the sign characteristics of ⁎-palindromic matrix polynomials

We study the eigendecompositions of para-Hermitian matrices H(z), that is, matrix-valued functions that are analytic and Hermitian on the unit circle S1⊂C. In particular, we fill existing gaps in the literature and prove the existence of a decomposition H(z)=U(z)D(z)U(z)P where, for all z∈S1, U(z) is unitary, U(z)P=U(z)⁎ is its conjugate transpose, and D(z) is real diagonal; moreover, U(z) and D(z) are analytic functions of w=z1/N for some positive integer N, and U(z)P is the so-called para-Hermitian conjugate of U(z). This generalizes the celebrated theorem of Rellich for matrix-valued functions that are analytic and Hermitian on the real line. We also show that there exists a decomposition H(z)=V(z)C(z)V(z)P where C(z) is pseudo-circulant, V(z) is unitary and both are analytic in z. We argue that, in fact, a version of Rellich's theorem can be stated for matrix-valued function that are analytic and Hermitian on any line or any circle on the complex plane. Moreover, we extend these results to para-Hermitian matrices whose entries are Puiseux series (that is, on the unit circle they are analytic in w but possibly not in z). Finally, we discuss the implications of our results on the singular value decomposition of a matrix whose entries are S1-analytic functions of w, and on the sign characteristics associated with unimodular eigenvalues of ⁎-palindromic matrix polynomials.

The second immanantal polynomials of Laplacian matrices of unicyclic graphs

Let M=(aij) be a square matrix of order n. The second immanant of M isd2(M)=∑σ∈Snχ(σ)∏t=1natσ(t),where χ is the irreducible character of Sn corresponding to the partition (2,1n−2). Let G be a graph. Denote by L(G) the laplacian matrix of G. The polynomial ϕ(G,x)=d2(xI−L(G))=∑k=0n(−1)kck(G)xn−k is called the second immanantal polynomial of G, where ck(G) is the k-th coeficient of ϕ(G,x). The coefficient ck(G) admit various algebraic and topological interpretations for a graph. Merris first considered the coefficient cn−1(G), and he proposed an open problem: given n and m, can the graph G be determined for which cn−1(G) is a minimum or maximum? In this article, we investigate the problem, and we construct six graph operations each of which changes the value of cn−1(G). Using these graph operations, we determine the bound of cn−1(G) of unicyclic graphs. In particular, we also determined the corresponding extremal graphs.

An efficient algorithm for constructing reversible quasi-cyclic codes via Chinese remainder theorem

Regarding quasi-cyclic codes as certain polynomial matrices, we show that all reversible quasi-cyclic codes are decomposed into reversible linear codes of shorter lengths corresponding to the coprime divisors of the polynomials with the form of one minus x to the power of m. This decomposition brings us an efficient method to construct reversible quasi-cyclic codes. We also investigate the reversibility and the self-duality of the linear codes corresponding to the coprime divisors of the polynomials. Specializing to the cases where the number of cyclic sections is not more than two, we give necessary and sufficient conditions for the divisors of the polynomials for which the self-dual codes are reversible and the reversible codes of half-length-dimension are self-dual. Our theorems are utilized to search reversible self-dual quasi-cyclic codes with two cyclic sections over binary and quaternary fields of lengths up to seventy and thirty-six, respectively, together with the maximums of their minimum weights.

Stability of matrix polynomials in one and several variables

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.

Determinantal polynomials and the base polynomial of a square matrix over a finite field

PurposeThe author studies forms over finite fields obtained as the determinant of Hermitian matrices and use these determinatal forms to define and study the base polynomial of a square matrix over a finite field.Design/methodology/approachThe authors give full proofs for the new results, quoting previous works by other authors in the proofs. In the introduction, the authors quoted related references.FindingsThe authors get a few theorems, mainly describing some monic polynomial arising as a base polynomial of a square matrix.Originality/valueAs far as the author knows, all the results are new, and the approach is also new.

Модальный синтез регуляторов полиномиальным методом для систем автоматического управления с избыточной размерностью вектора управления

When using polynomial methods of synthesis of multichannel regulators, there is a need for polynomial matrix calculus. However, when using this method, objects with the number of output channels equal to the number of input channels are mainly considered. This is necessary for the convenience of solving a system of linear algebraic equations in matrix polynomial computation. At the same time, a fairly large number of real technical systems have an unequal number of input and output channels. At the same time, the issue of the synthesis of generators by the polynomial method for multichannel objects with an unequal number of input and output impacts has not been worked out sufficiently deeply. One of the special cases of such systems can be considered systems with excessive dimension of the control vector. Within the framework of this work, examples of such systems and the purposes of their use are given. An illustrative example of a linear model of an unstable object with three channels for input action and two channels for output action is given. It is necessary to achieve certain quality indicators of the output vector quantity, while the control is carried out in the feedback of the system and is summed up with the input effect. The simplicity of the system under consideration is connected with the convenience of demonstrating on this example a modal synthesis method using a polynomial matrix decomposition of the transfer functions of an object and a controller for such a class of objects. In accordance with the recommendations presented in the algorithm for synthesizing regulators for objects with a non-square matrix transfer function, to solve the problem of synthesizing a regulator for systems with an excessive dimension of the control vector, the transfer function of the control object is represented as a right polynomial matrix decomposition, and the regulator is represented as a left one. During the demonstration of the example of the algorithm, some clarifications and edits are proposed in it.

Полиномиальный матричный метод синтеза регуляторов для объектов с запаздыванием управляющего сигнала

In this paper, we consider the synthesis of control systems for objects with a control signal delay by a polynomial matrix method, which is used to locate the poles and, if possible, zeros in the required position. The controller is calculated from the output, i.e. only from the measured values in the plant, which is an advantage over other modal direction synthesis methods where a state vector must be used. It is proposed to approximate the delay link with a Padé series with a limited number of terms, thus obtaining transfer functions of the first and second orders. The desired characteristic polynomial of the closed system is chosen so that it contains the denominators of the approximation transfer functions, which will keep their poles in the closed system. The polynomial synthesis method makes it possible to calculate multichannel controllers both for objects with multiple inputs and multiple outputs (multi input - multi output, MIMO) and for objects with one input and several outputs (single input - multi output, SIMO). The latter include a DC motor with independent excitation, where the armature current and rotor speed are outputs, and the control signal applied to the semiconductor converter is the input. In this work, the control signal is formed with a delay exceeding the time of the transient process of the engine, which significantly affects its dynamics. By applying the proposed approach, it was possible to synthesize a rotor speed control system that is resistant to changes in the delay time in a fairly wide range.

Stabilization of Linear Time-Invariant Systems by State-Derivative Feedback

In this paper is studied the stabilization problem by state-derivative feedback for linear time-invariant continuous-time systems. In particular, explicit necessary and sufficient conditions are established for the stability of a closed-loop system, obtained by state-derivative feedback from the given linear time-invariant continuous-time system. Furthermore a procedure is given for the computation of stabilizing state-derivative feedback. Our approach is based on properties of real and polynomial matrices.

Algebro-geometric quasi-periodic solutions to the Satsuma–Hirota hierarchy

In this paper, we study the theory of the resulting tetragonal curve and derive three kinds of Abel differentials, Baker–Akhiezer function and meromorphic function and so on, from which a systematic method is developed to construct algebro-geometric quasi-periodic solutions of soliton equations associated with the 4 × 4 matrix spectral problems. As an illustrative example, the Satsuma–Hirota hierarchy related to the 4 × 4 matrix spectral problem is obtained by utilizing the Lenard recursion equation and zero-curvature equation. Based on the characteristic polynomial of Lax matrix for the Satsuma–Hirota hierarchy, we introduce a tetragonal curve Kg of genus g and study the asymptotic properties of the Baker–Akhiezer function ψ1 and the meromorphic function ϕ near the infinite points on the tetragonal curve. The straightening out of various flows is exactly given through the Abel map and Abel–Jacobi coordinates. Using the theory of the tetragonal curve and the properties of the three kinds of Abel differentials, we obtain the explicit Riemann theta function representations of the Baker–Akhiezer function and the meromorphic function, and in particular, that of solutions for the entire Satsuma–Hirota hierarchy.

Admissible transformation approach to Roesser state‐space model realization of singular multidimensional systems

AbstractThe problem of the singular Roesser (state‐space) model realization of non‐causal multivariate transfer (function) matrices is investigated. Specifically, the notion of so‐called admissible transformation is introduced, which allows to introduce and investigate a novel invariant matrix relationship on polynomial matrices. The singular multidimensional Roesser model (SNDRM) realization is transformed to the problem of how to reduce an initial polynomial matrix associated with the given transfer matrix into the objective polynomial matrix with a specific structure. Construction procedures are developed to obtain SNDRMs by applying admissible transformations on the initial polynomial matrix. Moreover, a separated standard form is introduced for an SNDRM so that the proposed approach can effectively treat both the left and right matrix fractional descriptions of a non‐causal multivariate transfer matrix in a unified manner. Non‐trivial examples are provided to illustrate the details and effectiveness of the proposed approach.

Multistep Methods for Second-Order Differential-Algebraic Equations

We consider the initial value problem for linear systems of second-order ordinary differential equations with an identically singular matrix multiplying the principal part. Sufficient conditions for the existence of a unique solution are given in terms of matrix polynomials. For such problems, multistep difference schemes are proposed. An analysis of their stability and calculations of a model example are carried out.

A CLT for the characteristic polynomial of random Jacobi matrices, and the G$$\beta $$E

We prove a central limit theorem for the real part of the logarithm of the characteristic polynomial of random Jacobi matrices. Our results cover the G $$\beta $$ E models for $$\beta >0$$ .

Bilinear matrix inequalities and polynomials in several freely noncommuting variables

Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.

On the quasi-orthogonality and Hahn-classical d-orthogonal polynomials

This paper focuses on the quasi-orthogonality between sequences of -orthogonal polynomials and their characterization in terms of linear combinations as well as the left product of vector of linear forms by matrix polynomial. Therefore, structure relations for Hahn-classical -orthogonal polynomials are provided.

On the characteristic polynomial and energy of Hermitian quasi-Laplacian matrix of mixed graphs

A mixed graph is a graph whose edge set consists of both oriented and unoriented edges. The Hermitian-adjacency matrix of an [Formula: see text]-vertex mixed graph is a square matrix [Formula: see text] of order [Formula: see text], where [Formula: see text] if there is an arc from [Formula: see text] to [Formula: see text] and [Formula: see text] if there is an edge between [Formula: see text] and [Formula: see text], and [Formula: see text] otherwise. Let [Formula: see text] be a diagonal matrix, where [Formula: see text] is the degree of [Formula: see text] in the underlying graph of [Formula: see text]. The matrices [Formula: see text] and [Formula: see text] are, respectively, the Hermitian Laplacian and Hermitian quasi-Laplacian matrix of the mixed graph [Formula: see text]. In this paper, we first found coefficients of the characteristic polynomial of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph [Formula: see text]. Second, we discussed relationship between the spectra of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph [Formula: see text].

On General Factorization Problems of n-D Polynomial Matrices

Different from previous perspective, this brief studies relationships between the modules <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho ({\mathbf {F}}_{1})$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho ({\mathbf {F}})$ </tex-math></inline-formula> generated by the rows of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbf {F}}_{1}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbf {F}}$ </tex-math></inline-formula> respectively under the assumption of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbf {F}}$ </tex-math></inline-formula> admitting a general factorization <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbf {F}}={\mathbf {G}}_{1}{\mathbf {F}}_{1}$ </tex-math></inline-formula> . Two necessary and sufficient conditions for the equivalence of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho ({\mathbf {F}}_{1})$ </tex-math></inline-formula> and quotient modules of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho ({\mathbf {F}})$ </tex-math></inline-formula> with respect to some polynomials are obtained for two different situations. As a by-product, some general factorizations of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbf {F}}$ </tex-math></inline-formula> by computing related quotient modules can be got.

Stabilization of multivariable linear time-invariant systems by proportional-plus-derivative state feedback

Abstract This paper is devoted to the study of the problem of stabilization by proportional-plus-derivative state feedback for multivariable linear time-invariant systems. In particular, explicit necessary and sufficient conditions are established for the stability of a closed-loop system obtained by proportional-plus-derivative state feedback from the given multivariable linear time-invariant system. A procedure is given for the computation of stabilizing proportional-plus-derivative state feedback. Our approach is based on properties of real and polynomial matrices.