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This article is focused on the design of an output tracking control scheme for a class of continuous-time periodic piecewise time-varying systems (PPTVSs) with actuator saturation and nonlinear perturbations. The energy-to-peak tracking performance is studied based on an equivalent condition on the definiteness property of matrix polynomials. Considering the actuator saturation and nonlinear perturbation, matrix polynomial-based sufficient conditions are derived through the Lyapunov method using periodic matrix functions. From a perspective of subinterval segmentation aimed at PPTVSs, the proposed conditions can achieve less conservatism for tracking the output of a periodic time-varying reference system, while the controller gains can be computed using convex optimization. Moreover, a heuristic algorithm is constructed to simultaneously guarantee the closed-loop state convergence and the output tracking performance. The reduction in conservatism and the effectiveness of algorithm are demonstrated by illustrative case studies.

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 x n matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure. This work has been supported by the Agencia Estatal de Investigación of Spain through grants PID2019-106362GB-I00 MCIN/ AEI/10.13039/501100011033/ and MTM2017-90682-REDT, and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).

In this article, we generalize previously reported results for linear, time-invariant, stabilizable multivariable systems described by a strictly proper transfer function matrix $P(s)$ with number of outputs greater than or equal to the number of inputs. By making use of a special kind of a left generalized inverse $P(s)_{\alpha }^{\oplus }$ of $P(s)$ , we define and examine the equivalent relation $\mathcal {R}$ relating $P(s)$ with the members of the equivalence class $[P(s)]_{R}$ of the closed loop-transfer function matrices $P_{C}(s)$ obtainable from $P(s)$ by the use of a proper compensator $C(s)$ in the feedback path. For $\mathcal {R}$ , we establish a set of complete invariants and a canonical form. These results give rise to a simple algorithmic procedure for the computation of proper internally stabilizing and denominator assigning compensators $C(s)$ for the class of plants with $p=m$ and having no zeros in the closed right half complex plane: $\mathbb {C}^{+}$ and in the case when $p>m$ plants characterized by right polynomial matrix fraction descriptions with a polynomial matrix numerator having at least one subset of $m$ rows that give rise to a nonsingular polynomial matrix with no zeros in $\mathbb {C}^{+}$ .

A new general approach for solving matrix polynomial equations of arbitrary order with matrix or vector unknowns is proposed in the work with the use of nested continued fractions.

In this paper, the output reachable set synthesis problem for periodic piecewise time-varying systems has been addressed for the first time. Generalized periodic time-varying conditions of the closed-loop system to guarantee the system output bounded by a collection of finite ellipsoids are developed. The decoupling of time-varying observer and controller gains is achieved by a technical lemma . Moreover, by using time-varying matrix functions in multiple linear-interpolative forms and the relaxed quadratic matrix polynomial definiteness results, sufficient condition is developed to determine the periodic time-varying observer and controller with reduced conservatism. A tractable iterative algorithm is proposed to obtain the periodic time-varying observer and controller gains. A numerical example is presented to illustrate the validity of proposed methods.

We construct new "standard modules" for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson-Schensted-Knuth correspondence for Zelevinsky's multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis., Comment: With an appendix by Mark Shimozono

This paper proposes a comprehensive solution to persistent problems faced by nonlinear observer design via convex optimization techniques, namely, the construction of an exact error model and the handling of unmeasurable premises. Instead of using the differential mean value theorem, it is shown that the error system is always exactly amenable, via algebraic rearrangements, to an explicit matrix polynomial form of the gradient, where the error signal appears factorized at the rightmost side, thus allowing Lyapunov-based derivation of conditions in the form of linear matrix inequalities or sum of squares. An effective exploitation of the nonlinear characteristics of the observer while naturally decoupling available and non-available signals, is achieved via exact Takagi-Sugeno tensor-product models or polynomial rewriting of the error system, not the plant; generalizations such as H ∞ disturbance rejection or decay rate are straightforward. In contrast with recent approaches, the output is not required to be linear nor the systems involved to be affine-in-control. Illustrative examples are provided that overcome recent approaches on the subject.

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational...

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.

In this paper, the theory and effectiveness of two polynomial approaches are compared in the analysis of L 2 - L ∞ and H ∞ performance for a type of periodic piecewise polynomial systems, where the time-varying subsystems can be characterized in Bernstein polynomials. Using the Bernstein polynomial-based lemma and the existing lemma concerning the negativity/positivity of matrix polynomial functions, sufficient conditions are established in tractable forms aimed at the global asymptotic stability and performance analysis. Four cases of optimization constraints are considered based on the proposed conditions. The performance indices obtained via the four cases are compared through a numerical example, and the lower conservatism achieved by the proposed Bernstein polynomial approach is demonstrated.

A new way to formulate the notions of minimal basis and minimal indices is developed in this paper, based on the concept of a filtration of a vector space. The goal is to provide useful new tools for working with these important concepts, as well as to gain deeper insight into their fundamental nature. This approach also readily reveals a strong minimality property of minimal indices, from which follows a characterization of the vector polynomial bases in rational vector spaces. The effectiveness of this new formulation is further illustrated by proving several fundamental properties: the invariance of the minimal indices of a matrix polynomial under field extension, the direct sum property of minimal indices, the polynomial linear combination property, and the predictable degree property.

The structured partial quadratic inverse eigenvalue problem (SPQIEP) is to construct the structured quadratic matrix polynomial using the partial eigendata. The structures arising in physical appli...

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.

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.

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.

This paper investigates the problem of non-fragile H ∞ control for periodic piecewise time-varying systems. Based on a Lyapunov function with continuous time-varying Lyapunov matrix polynomial, and...

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. Three 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 a special matrix polynomial. The results are illustrated with various examples and numerical simulations., Comment: 32 pages, 6 figures

International audience; We rely on aggregate separation bounds for univariate polynomials to introduce novel worst-case separation bounds for the isolated roots of zero-dimensional, positive-dimensional, and overde- termined polynomial systems. We exploit the structure of the given system, as well as bounds on the height of the sparse (or toric) resultant, by means of mixed volume, thus establishing adaptive bounds. Our bounds improve upon Canny’s Gap theorem [9]. Moreover, they exploit sparseness and they apply without any assumptions on the input polynomial system. To evaluate the quality of the bounds, we present polynomial systems whose root separation is asymptotically not far from our bounds.We apply our bounds to three problems. First, we use them to estimate the bitsize of the eigenvalues and eigenvectors of an integer matrix; thus we provide a new proof that the problem has polynomial bit complexity. Second, we bound the value of a positive polynomial over the simplex: we improve by at least one order of magnitude upon all existing bounds. Finally, we asymptotically bound the number of steps of any purely subdivision-based algorithm that isolates all real roots of a polynomial system.

In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for tensors corresponding to T-product. Furthermore, we discuss on the solution of least-squares problem associated with a tensor equation using Tensor Singular Value Decomposition (TSVD). Motivated by the effectiveness of some proposed vector extrapolation methods in earlier papers, we describe how an extrapolation technique can be also implemented on the sequence of tensors produced by truncated TSVD (TTSVD) for solving possibly ill-posed tensor equations.

Palindromic matrix polynomials arise in many applications. Structure-preserving linearizations of palindromic matrix polynomials have been proposed in the literature so as to preserve the spectral symmetry in the eigenvalues. We present a new family of palindromic strong linearizations of a palindromic matrix polynomial of odd degree. A salient feature of the new family is that it allows the construction of banded palindromic linearizations of block-bandwidth k + 1 for any k = 0 : m − 2, where m is the degree of the palindromic matrix polynomial. Low bandwidth palindromic pencils may be useful for numerical computations. Our construction of the new family is based on Fiedler companion matrices associated with matrix polynomials and the construction is operation-free. Moreover, the new family of palindromic pencils allows operation-free recovery of eigenvectors and minimal bases, and an easy recovery of minimal indices of matrix polynomials from those of the palindromic linearizations. We also present an operation-free algorithm for construction of palindromic pencils belonging to the new family.

A 8-cube model of the fully nonrigid water octamer is considered within the 8-dimensional hyperoctahedral wreath product group with 10,321,920 operations and 185 irreducible representations by employing computational and mathematical techniques. For the two lowest-lying isomers of (H2 O)8 with D2d and S4 symmetries of a rigid (H2 O)8 , correlation tables and nuclear spin statistics are constructed for the tunneling splittings of the rotational levels are computed by a computational matrix polynomial generating function technique combined with Mobius inversion, and the relationship to the 8-cube multinomials are pointed out. Multinomial generating functions combined with the induced representation techniques are employed to compute and construct the nuclear spin species, nuclear spin statistical weights and tunneling splittings of rovibronic levels. We have also computed the spin statistical weights and tunneling splittings of the rotational levels for a semirigid water octamer within the wreath product Oh [S2 ] consisting of 12,288 operations.

We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite field. As an application of our results we give a new proof of a theorem of Chen and Tseng which answers a question of Niederreiter on splitting subspaces. We use our results to affirmatively resolve a conjecture on the probability that a matrix polynomial is unimodular., 14 pages

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.

Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest matrix polynomial that is algebraically singular with a prescribed lower bound on the dimension given in a previous paper by the authors. In this paper we prove that such lower rank matrices at minimal distance always exist, satisfy regularity conditions, and are all isolated and surrounded by a basin of attraction of non-minimal solutions. In addition, we present an iterative algorithm which, on given input sufficiently close to a rank-at-most matrix, produces that matrix. The algorithm is efficient and is proven to converge quadratically given a sufficiently good starting point. An implementation demonstrates the effectiveness and numerical robustness of our algorithm in practice., 31 Pages

This paper proposes a design method of model predictive control (MPC) for multi-input multi-output (MIMO) plants with time-delay by using an interactor matrix and a sequential procedure to solve th...

This paper studies the problem of finite-time non-fragile extended dissipative control for the periodic piecewise time-varying systems. First, based on the time-varying Lyapunov matrix and the matrix polynomial condition, a sufficient condition of finite-time boundedness for periodic piecewise time-varying subsystem is derived, and the finite-time extended dissipative performance is then analyzed. Furthermore, considering two types of time-varying controller perturbations, the non-fragile controllers are developed, which can guarantee the finite-time boundedness of periodic piecewise time-varying systems and satisfy the extended dissipative performance at the same time. Finally, numerical examples demonstrate the effectiveness of the proposed methods.

This paper investigates the secure synchronization control problem for a class of cyber-physical systems &#x0028 CPSs &#x0029 with unknown system matrices and intermittent denial-of-service &#x0028 DoS &#x0029 attacks. For the attack free case, an optimal control law consisting of a feedback control and a compensated feedforward control is proposed to achieve the synchronization, and the feedback control gain matrix is learned by iteratively solving an algebraic Riccati equation &#x0028 ARE &#x0029 . For considering the attack cases, it is difficult to perform the stability analysis of the synchronization errors by using the existing Lyapunov function method due to the presence of unknown system matrices. In order to overcome this difficulty, a matrix polynomial replacement method is given and it is shown that, the proposed optimal control law can still guarantee the asymptotical convergence of synchronization errors if two inequality conditions related with the DoS attacks hold. Finally, two examples are given to illustrate the effectiveness of the proposed approaches.

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.

We present a generalization of Householder sets for matrix polynomials. After defining these sets, we analyze their topological and algebraic properties, which include containing all of the eigenvalues of a given matrix polynomial. Then, we use instances of these sets to derive the Gersgorin set, weighted Gersgorin set, and weighted pseudospectra of a matrix polynomial. Finally, we show that Householder sets are intimately connected to the Bauer-Fike theorem by using these sets to derive Bauer-Fike-type bounds for matrix polynomials.

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.

The aim of this paper is to derive a new family of multi-block boundary value methods based on multi-block generalized Adams methods for numerical solution of stiff problems. Generally, boundary value methods easily circumvents the Dahlquist stability barrier in linear multistep formulas. This leads to a new stability framework for multi-block boundary value methods (MB $$_2$$ VMs) with the aid of the Wiener–Hopf factorization of a matrix polynomial, which generalizes the stability of initial multi-block methods in Chu and Hamilton (SIAM J Sci Stat Comput 8:342–535, 1987). The condition number of the arising Toeplitz matrix for the MB $$_2$$ VMs are very small, and in general are bounded. The MB $$_2$$ VMs are new classes of very large scale integration methods for the numerical solution of differential equations which output multi-block of solutions on application. Numerical experiments are presented and compared with some existing ones.

In this note, given a matrix A∈Cn×n (or a general matrix polynomial P(z), z∈C) and an arbitrary scalar λ0∈C, we show how to define a sequence μkk∈N which converges to some element of its spectrum. The scalar λ0 serves as initial term (μ0=λ0), while additional terms are constructed through a recursive procedure, exploiting the fact that each term μk of this sequence is in fact a point lying on the boundary curve of some pseudospectral set of A (or P(z)). Then, the next term in the sequence is detected in the direction which is normal to this curve at the point μk. Repeating the construction for additional initial points, it is possible to approximate peripheral eigenvalues, localize the spectrum and even obtain spectral enclosures. Hence, as a by-product of our method, a computationally cheap procedure for approximate pseudospectra computations emerges. An advantage of the proposed approach is that it does not make any assumptions on the location of the spectrum. The fact that all computations are performed on some dynamically chosen locations on the complex plane which converge to the eigenvalues, rather than on a large number of predefined points on a rigid grid, can be used to accelerate conventional grid algorithms. Parallel implementation of the method or use in conjunction with randomization techniques can lead to further computational savings when applied to large-scale matrices.

In the processing of signals defined over graph domains, it is highly desirable to have algorithms that can be implemented in a decentralised manner, whereby each node only needs to exchange information within a localized subgraph of nodes. The most common example is when the subgraph consist of immediate neighbors and this allows for distributed processing. Many graph signal operators, in their original forms, are not amenable to distributed processing. To overcome this deficiency, a polynomial approximation is usually applied to the original operator to yield a distributed operator which is a matrix polynomial of the graph shift matrix. This approach however is only applicable when the original operator is a function of the graph shift matrix. In this paper, we propose a generalized approach to approximate graph signal operators that are not necessary functions of the shift matrix. The key idea here is to restrict the approximated matrix inverse to have small geodesic-width so that multiplication with the small geodesic-width matrix can be implemented in a decentralised manner. Furthermore, to increase the accuracy of the approximated operator, an iterative algorithm which also has the decentralised property and low computational complexity, is proposed. We apply the proposed approach to signal inpainting and signal reconstruction in filter banks. Numerical results verify the effectiveness of the proposed algorithm. The proposed algorithms can also be used to efficiently solve linear system of equations.

International audience; Assuming the variety of a polynomial set is invariant under a group action, we construct a set of invariants that define the same variety. Our construction can be seen as a generalization of the previously known construction for finite groups. The result though has to be understood outside an invariant variety which is independent of the polynomial set considered. We introduce the symmetrizations of a polynomial that are polynomials in a generating set of rational invariants. The generating set of rational invariants and the symmetrizations are constructed w.r.t. a section to the orbits of the group action.

In this study, the stability and stabilisation issues of a class of periodic piecewise linear time-varying systems with time delay are studied. Based on a lemma with respects to the negative definite matrix polynomial, a sufficient delay-dependent exponential stability condition is obtained via constructing a Lyapunov–Krasovskii functional with periodic matrix functions. Then, for two types of norm-bounded uncertainties of time-varying controller gains, the non-fragile stabilising controllers under the multiplicative and additive gain perturbations are designed to stabilise the closed-loop system. The obtained controller gains could be solved with linear matrix inequalities directly instead of using the iterative algorithm. That is valued in engineering applications. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.

In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques.

In several situations, mainly involving a self-adjoint set of unitary generators of a $C^*$-algebra, we show that any matrix polynomial in the generators and the unit that is in the open unit ball can be written as a product of matrix polynomials of degree 1 also in the open unit ball., Comment: v2 minor improvements mainly in the introduction

This paper presents a method for designing inputs to identify linear continuous-time multiple-input multiple-output (MIMO) systems. The goal here is to design a T-optimal band-limited spectrum satisfying certain input/output power constraints. The input power spectral density matrix is parametrized as the product $$\phi_u(\text{j}\omega)=\frac{1}{2}H(\text{j}\omega)H^\text{H}(\text{j}\omega)$$ , where H(jω) is a matrix polynomial. This parametrization transforms the T-optimal cost function and the constraints into a quadratically constrained quadratic program (QCQP). The QCQP turns out to be a non-convex semidefinite program with a rank one constraint. A convex relaxation of the problem is first solved. A rank one solution is constructed from the solution to the relaxed problem. This relaxation admits no gap between its solution and the original non-convex QCQP problem. The constructed rank one solution leads to a spectrum that is optimal. The proposed input design methodology is experimentally validated on a cantilever beam bonded with piezoelectric plates for sensing and actuation. Subspace identification algorithm is used to estimate the system from the input-output data.

Linearization is a widely used method for solving polynomial eigenproblems in which a matrix polynomial is transformed to a matrix pencil of larger size. Fiedler pencils, generalized Fiedler pencils, Fiedler pencils with repetition (FPRs) and generalized Fiedler pencils with repetition (GFPRs) are well known classes of strong linearizations of matrix polynomials. The class GFPRs is also a rich source of structure-preserving strong linearizations of structured matrix polynomials. The recovery of eigenvectors, minimal bases and minimal indices of matrix polynomials from those of the linearizations is an important task. It is well known that eigenvectors and minimal bases of matrix polynomials can be easily recovered from those of the Fiedler and generalized Fiedler pencils. By contrast, barring a small subclass of FPRs, the recovery of eigenvectors and minimal bases of matrix polynomials from those of the FPRs and GFPRs is still an open problem. The purpose of this paper is to fill this gap. We consider the class of GFPRs, which subsumes the class of FPRs, and describe the recovery of eigenvectors, minimal bases and minimal indices of matrix polynomials from those of the GFPRs and show that the recovery is operation-free.

Let G ( λ ) be an n × n rational matrix. By considering a minimal realization of G ( λ ) , Fiedler-like pencils (such as Fiedler pencils, generalized Fiedler pencils and Fiedler pencils with repetition) of G ( λ ) have been proposed recently which are shown to be linearizations of G ( λ ) . We show that the Fiedler-like pencils allow operation-free recovery of eigenvectors and minimal bases of G ( λ ) , that is, eigenvectors and minimal bases of G ( λ ) can be recovered from those of the Fiedler-like pencils of G ( λ ) without performing any arithmetic operations. Further, we show that the minimal indices of G ( λ ) can be easily recovered from those of the Fiedler-like pencils of G ( λ ) . We also show that a Fiedler pencil with repetition of G ( λ ) can be constructed directly from that of a matrix polynomial.

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.

Polynomial methods for synthesizing linear regulators for automatic control systems with linear objects, proposed by a number of authors, including Chen, Kailath, Gaiduk, and others, along with methods of synthesis in the state space, are becoming increasingly widespread. The synthesis of multichannel regulators caused by the need to use the matrix polynomial calculus is of a special difficulty, which is aggravated by a significant increase in the dimension of the matrices during the transition from polynomial matrices to numeric ones, in which Sylvester matrices are used. Herewith, it is necessary to take into account the requirements of controllability and observability, leading to the need to check for the presence of identical roots in polynomial matrices corresponding to the numerator and denominator of the object. This leads to the requirement of a relatively prime matrix polynomial fraction, which can be significantly weakened if it is possible to include in the desired characteristic matrix of the system some zeros and poles of the object located far to the left of the imaginary axis. In calculations using numerical matrices and, consequently, using Sylvester matrices, the latter degenerate due to the lowering of the rank, which complicates the calculations. The research continues to study polynomial synthesis of multichannel regulators based on the results obtained by Chen and other researchers and presents an algorithm for the synthesis of regulators, the feature of which is the possibility of introducing additional so-called free parameters that allow additional requirements for the automatic control system. The free parameters allow to obtain strictly proper regulators, along with the proper regulators.

This paper is devoted to the application aspects of homomorphic cryptography. It provides a description of a fully homomorphic matrix polynomial-based encryption scheme. It also gives the results of practical comparison of fully homomorphic encryption schemes. We consider some special cases of homomorphic encryption allowing computations of a limited number of functions.

In this paper, we investigate the sample size requirement for exact recovery of a high order tensor of low rank from a subset of its entries. We show that a gradient descent algorithm with initial value obtained from a spectral method can, in particular, reconstruct a ${d\times d\times d}$ tensor of multilinear ranks $(r,r,r)$ with high probability from as few as $O(r^{7/2}d^{3/2}\log^{7/2}d+r^7d\log^6d)$ entries. In the case when the ranks $r=O(1)$, our sample size requirement matches those for nuclear norm minimization (Yuan and Zhang, 2016a), or alternating least squares assuming orthogonal decomposability (Jain and Oh, 2014). Unlike these earlier approaches, however, our method is efficient to compute, easy to implement, and does not impose extra structures on the tensor. Numerical results are presented to further demonstrate the merits of the proposed approach., 56 pages, 4 figures

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.

The combinatorics of walks on a graph is a key topic in network science. Here we study a special class of walks on directed graphs. We combine two features that have previously been considered in isolation. We consider alternating walks, which form the basis of algorithms for hub/authority detection and for discovering directed bipartite substructure. Within this class, we restrict to non-backtracking walks, since this constraint has been seen to offer advantages in related contexts. We derive a recursive formula for counting the total number of non-backtracking alternating walks of a given length, leading to an expression for any associated power series expansion. We discuss computational issues for the widely used cases of resolvent and exponential series, showing that non-backtracking can be incorporated at very little extra cost. We also derive an appropriate asymptotic limit which gives a parameter-free, spectral analogue. We perform tests on an artificial data set in order to quantify the advantages of the new methodology. We also show that the removal of backtracking allows us to identify larger bipartite subgraphs within an anatomical connectivity network from neuroscience.

Let $P : \Omega \subset {\mathbb C} \rightarrow {\mathbb C}^{n\times n}$ be given by $P(\lambda) :=\sum^m_{j=0}A_j\phi_j(\lambda),$ where $ \phi_j : \Omega \rightarrow {\mathbb C}$ for $j=0, 1, \ld...