Companion Matrix Research Articles

A note on Johansen's rank conditions and the Jordan form of a matrix

This note presents insights on the Jordan structure of a matrix which are derived from an extension of the and conditions in Johansen (1996). It is first observed that these conditions not only characterize, as it is well known, the size (1 or 2) of the largest Jordan block in the Jordan form of the companion matrix but more generally the geometric multiplicities, the algebraic multiplicities and the whole Jordan structure for eigenvalues of index 1 or 2. In the context of the Granger representation theorem, this means that the Johansen rank conditions do more than determine the order of integration of the process. It is then shown that an extension of these conditions leads to the characterization of the Jordan structure of any matrix.

On Numerical Radius Inequalities for Hilbert Space Operators

This paper aims to find special cases of some inequalities for numerical radii and spectral radii of a bounded linear operator on a Hil-bert space, we focus on numerical radii inequalities for restricted linear operators on complex Hil-bert spaces for the case of one and two operators, and study the numerical range of an operator K on a complex Hil-bert space H, after that we present some inequalities for numerical radii and spectral radii and studied it to find new results. At the end of this paper we find several inequalities for numerical radii by using the spectral norm, this study is necessary to find other bound for zeros of polynomials and this study is necessary to find new bounds for the zeros of polynomials by a playing the new results to the companion matrix.

A new quadra polynomial sequence

In this paper, we define a new quadra polynomial sequence by using Özkoç numbers as the coefficients. Then, we derive some properties for this polynomial sequence by the help of Fibonacci and Pell polynomials. Additionally, we attempt to define the companion matrix of this polynomial sequence.

Subresultants of several univariate polynomials in Newton basis

In this paper, we consider the problem of formulating the subresultant polynomials for several univariate polynomials in Newton basis. It is required that the resulting subresultant polynomials be expressed in the same Newton basis as that used in the input polynomials. To solve the problem, we devise a particular matrix with the help of the companion matrix of a polynomial in Newton basis. Meanwhile, the concept of determinant polynomial in power basis for formulating subresultant polynomials is extended to that in Newton basis. It is proved that the generalized determinant polynomial of the specially designed matrix provides a new formula for the subresultant polynomial in Newton basis, which is equivalent to the subresultant polynomial in power basis. Furthermore, we show an application of the new formula in devising a basis-preserving method for computing the gcd of several Newton polynomials.

Fitting of Coupled Potential Energy Surfaces via Discovery of Companion Matrices by Machine Intelligence.

Fitting coupled potential energy surfaces is a critical step in simulating electronically nonadiabatic chemical reactions and energy transfer processes. Analytic representation of coupled potential energy surfaces enables one to perform detailed dynamics calculations. Traditionally, fitting is performed in a diabatic representation to avoid fitting the cuspidal ridges of coupled adiabatic potential energy surfaces at conical intersection seams. In this work, we provide an alternative approach by carrying out fitting in the adiabatic representation using a modified version of the Frobenius companion matrices, whose usage was first proposed by Opalka and Domcke. Their work involved minimizing the errors in fits of the characteristic polynomial coefficients (CPCs) and diagonalizing the resulting companion matrix, whose eigenvalues are adiabatic potential energies. We show, however, that this may lead to complex eigenvalues and spurious discontinuities. To alleviate this problem, we provide a new procedure for the automatic discovery of CPCs and the diagonalization of a companion matrix by using a special neural network architecture. The method effectively allows analytic representation of global coupled adiabatic potential energy surfaces and their gradients with only adiabatic energy input and without experience-based selection of a diabatization scheme. We demonstrate that the new procedure, called the companion matrix neural network (CMNN), is successful by showing applications to LiH, H3, phenol, and thiophenol.

Correction: Hoffman–Wielandt type inequality for block companion matrices of certain matrix polynomials

Correction: Hoffman–Wielandt type inequality for block companion matrices of certain matrix polynomials

Oil price, energy consumption and CO2 emissions as growth determinants: a PVAR system GMM approach

Purpose This study aims to revisit the empirical debate about the asymmetric relationship between oil prices, energy consumption, CO2 emissions and economic growth in a panel of 184 countries from 1981 to 2020. Design/methodology/approach A relatively new research method, the PVAR system GMM, is applied. Findings The outcome of the PVAR system GMM model at the group level in the study suggests that oil prices exert a positive but statistically insignificant effect on economic growth. Energy consumption is inversely related to economic growth but statistically significant, and the correlation between CO2 emissions and economic growth is negative but statistically insignificant. The Granger causality test indicates that oil prices, CO2 emissions, oil rents, energy consumption and savings jointly Granger-cause economic growth. A unidirectional causality runs from energy consumption, savings and economic growth to oil prices. At countries’ income grouping levels, oil prices, oil rent, CO2 emissions, energy consumption and savings jointly Granger-cause economic growth for the high-income and upper-middle-income countries groups only, while those variables did not jointly Granger-cause economic growth for the low-income and lower-middle-income countries groups. The modulus emanating from the eigenvalue stability condition with the roots of the companion matrix indicates that the model is stable. The results support the asymmetric impacts of oil prices on economic growth and aid policy formulation, particularly the cross-country disparities regarding the nexus between oil prices and growth. Originality/value From a methodological perspective, to the best of the author’s knowledge, the study is the first attempt to use the PVAR system GMM and such a large sample group of 184 economies in the post-COVID-19 era to examine the impacts of oil prices on countries’ growth while controlling for other crucial variables, which is noteworthy. Two, using the World Bank categorisation of countries according to income groups, the study adds another layer of contribution to the literature by decomposing the 184 sample economies into four income groups: high-income, low-income, upper-middle-income and lower-middle-income groups to investigate the potential for asymmetric effects of oil prices on growth, the first of its kind in the post-COVID-19 period.

Estimating Posterior Sensitivities with Application to Structural Analysis of Bayesian Vector Autoregressions

The inherent feature of Bayesian empirical analysis is the dependence of posterior inference on prior parameters, which researchers typically specify. However, quantifying the magnitude of this dependence remains difficult. This article extends Infinitesimal Perturbation Analysis, widely used in classical simulation, to compute asymptotically unbiased and consistent sensitivities of posterior statistics with respect to prior parameters from Markov chain Monte Carlo inference via Gibbs sampling. The method demonstrates the possibility of efficiently computing the complete set of prior sensitivities for a wide range of posterior statistics, alongside the estimation algorithm using Automatic Differentiation. The method’s application is exemplified in Bayesian Vector Autoregression analysis of fiscal policy in U.S. macroeconomic time series data. The analysis assesses the sensitivities of posterior estimates, including the Impulse response functions and Forecast error variance decompositions, to prior parameters under common Minnesota shrinkage priors. The findings illuminate the significant and intricate influence of prior specification on the posterior distribution. This effect is particularly notable in crucial posterior statistics, such as the substantial absolute eigenvalue of the companion matrix, ultimately shaping the structural analysis.

The companion section for classical groups

We use the companion matrix construction for [Formula: see text] to build canonical sections of the Chevalley map [Formula: see text] for classical groups [Formula: see text] as well as the group [Formula: see text]. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and [Formula: see text].

Associations between Climate Variability and Livestock Production in Botswana: A Vector Autoregression with Exogenous Variables (VARX) Analysis

The changing climate has a serious bearing on agriculture, particularly livestock production in Botswana. Therefore, studying the relationship between climate and livestock, which at present is largely missing, is necessary for the proper formulation of government policy and interventions. This is critical in promoting the adoption of relevant mitigation strategies by farmers, thereby increasing resilience. The aim of this research is to establish associations between climate variability and livestock production in Botswana at the national level. The paper employs time series data from 1970 to 2020 and the Vector Autoregression with Exogenous Variables (VARX) model for statistical analysis. The trend shows that both cattle and goat populations are decreasing. The VARX model results reveal that cattle and goat populations are negatively associated with increasing maximum temperatures. Cattle respond negatively to increased minimum temperatures as well, while goats tend to respond positively, implying that livestock species react differently to climatic conditions due to their distinct features. The results of the roots of the companion matrix for cattle and goat production meet the stability condition as all the eigenvalues lie inside the unit circle. The study recommends further intervention by the government to deal with increasing temperatures, thereby addressing the dwindling populations of goats and cattle, which have significant contributions to the household economies of smallholders and the national economy, respectively.

Robust adaptive observer-based control of blood glucose level for type 1 diabetic patient

In this paper, an adaptive controller is designed to regulate the blood glucose level of type 1 diabetes mellitus while not all states of the system are measurable and also its parameters are unknown. The main goal in the control of diabetes is to preserve blood glucose level within a safe rang by a suitable injecting insulin rate to the patient. Herein, it is achieved by measuring the blood glucose level and proposed an observer based adaptive control system. In the proposed method, firstly, the dynamic equations of nonlinear Bergman minimal model (BMM) are transformed into a companion form. Then an adaptive observer is presented to simultaneously estimate the state variables and the system’s parameters. Afterward, based on the designed observer and using a new meal simulation model, an adaptive control is presented to bring back the blood glucose level to its safe range. The overall stability of the developed adaptive control is established using the Lyapunov direct method. Simulation results have been performed to verify the effectiveness of the proposed approach in tracking the desired blood glucose.

Computation of the zeros of a quaternionic polynomial using matrix methods

In a recent paper, Ishfaq Dar (2024), worked on the problem of locating the zeros of quaternion polynomials by introducing various matrix techniques. In this paper, we use those newly developed matrix methods to locate the left eigenvalues of the quaternion companion matrix of the polynomial, which in turn yield various results concerning the location of the zeros of quaternionic polynomials including extensions of the results of Rubinstein and A. Aziz to the quaternionic setting as well.

NOW SHOWING: THE USE OF FILMS IN ADVANCED GERONTOLOGY COURSES

Abstract While the use of films to teach about aging and older adults is not new, current movies and the students who view them are ever-changing. So too are the learning goals and practices for incorporating media into the advanced gerontological classroom. The current undertaking considers the unique aspects of employing movies (including documentaries and international films) and builds on previous works (Cohen-Shalev & Marcus, 2007; Karasik & Hamon, 2014; Markson & Taylor 1993) with an eye toward the more advanced pedagogical goals of upper level undergraduate and graduate courses. This poster focuses on challenges associated with including film in both traditional and emerging classroom delivery formats (e.g., face to face, hybrid, on-line). A matrix of current aging-themed films is provided, along with strategies for successful applications in upper level and graduate courses. Karasik and Hamon (2014) explored the use of popular film in introductory gerontology courses, creating a detailed matrix of films spanning 1951 - 2012. In addition to film descriptions and ratings, the matrix identified content in the following topical areas in (ageism/stereotypes; cognitive impairment; death & dying; diversity; family relationships; health & wellness; sexuality & intimacy; and work & retirement). The current presentation builds on Karasik & Hamon’s work by developing a companion matrix of films including feature films from 2012- 2023 , as well as documentaries and international films across the same time period. In addition to the newly created companion matrix, there are examples of activities and assignments that incorporate movies into the gerontological classroom.

Enabling attractive-repulsive potentials in binary-collision-approximation monte-carlo codes for ion-surface interactions

Binary Collision Approximation (BCA) codes for ion-material interactions, such as SRIM, Tridyn, F-TRIDYN, and SDtrimSP, have historically been limited to screened Coulomb potentials even at low energies due to the difficulty in numerically solving the Distance of Closest Approach (DOCA) problem for attractive-repulsive potentials. Techniques such as direct n-body simulation or modifications to Newton’s method are either prohibitively costly or not guaranteed to work for all potentials. Advanced rootfinding techniques, such as companion matrix solvers, offer a solution. For many attractive-repulsive potentials, however, a companion matrix cannot be used directly, because there is no way to put the associated functions into a monomial basis form. A complementary technique is proxy rootfinding—by finding the best-fit polynomial approximant of a function, the zeros of the approximant can be guaranteed to be close to the zeros of the function. Using the Chebyshev basis and grid offers additional guarantees with regards to the quality of the approximation, the speed of convergence, and the avoidance of Runge’s phenomenon. By finding Chebyshev interpolants and using the Chebyshev-Frobenius companion matrix, the zeros of any real function on a bounded domain can be found. Here we show that using an Adaptive Chebyshev Proxy Rootfinder with Automatic Subdivision (ACPRAS) with appropriate scaling functions, numerical issues presented by attractive-repulsive potentials, including those of scale, can be handled. Using these techniques, we show that it is possible to include any physically reasonable interatomic potential in a BCA code, and to guarantee correctness of the resulting scattering angle calculations.

Linear codes invariant under cyclic endomorphisms

In this paper, we study linear codes invariant under a cyclic endomorphism [Formula: see text] called [Formula: see text]-cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix [Formula: see text] that we call [Formula: see text]-cyclic codes, and when [Formula: see text] is the companion matrix [Formula: see text] of a given nonzero polynomial [Formula: see text] we call them [Formula: see text]-cyclic codes. The similarity relation between matrices helps us find connections between [Formula: see text]-cyclic codes and [Formula: see text]-cyclic codes, where [Formula: see text] is the minimal polynomial of [Formula: see text] The class of [Formula: see text]-cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between [Formula: see text]-cyclic codes and ideals of the polynomial ring [Formula: see text] where [Formula: see text] is the minimal polynomial of [Formula: see text] This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the [Formula: see text]-dual of an [Formula: see text]-cyclic code is an [Formula: see text]-cyclic code, where [Formula: see text] is the adjoint matrix of [Formula: see text] with respect to [Formula: see text] and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes.

Representatives of similarity classes of matrices over PIDs corresponding to ideal classes

Abstract For a principal ideal domain $A$ , the Latimer–MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\textrm{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and the ideal classes of the order $A[x]/(f(x))$ . We prove that when $A[x]/(f(x))$ is maximal (i.e. integrally closed, i.e. a Dedekind domain), then every similarity class contains a representative that is, in a sense, close to being a companion matrix. The first step in the proof is to show that any similarity class corresponding to an ideal (not necessarily prime) of degree one contains a representative of the desired form. The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.

On universal realizability in the left half-plane

A list Λ = {λ1, λ2,..., λn} of complex numbers is said to be realizable if it is the spectrum of a nonnegative matrix. Λ is said to be universally realizable (UR) if it is realizable for each possible Jordan canonical form allowed by Λ. In this paper, using companion matrices and applying a procedure by Šmigoc, we provide sufficient conditions for the universal realizability of left half-plane spectra, that is, spectra Λ = {λ1,...,λn} with λ1 > 0, Re λi ≤ 0, i = 2, . . . , n. It is also shown how the effect of adding a negative real number to a not UR left half-plane list of complex numbers, makes the new list UR, and a family of left half-plane lists that are UR is characterized.

The Hitchhiker’s guide to longitudinal models: A primer on model selection for repeated-measures methods

Longitudinal data are becoming increasingly available in developmental neuroimaging. To maximize the promise of this wealth of information on how biology, behavior, and cognition change over time, there is a need to incorporate broad and rigorous training in longitudinal methods into the repertoire of developmental neuroscientists. Fortunately, these models have an incredibly rich tradition in the broader developmental sciences that we can draw from. Here, we provide a primer on longitudinal models, written in a beginner-friendly (and slightly irreverent) manner, with a particular focus on selecting among different modeling frameworks (e.g., multilevel versus latent curve models) to build the theoretical model of development a researcher wishes to test. Our aims are three-fold: (1) lay out a heuristic framework for longitudinal model selection, (2) build a repository of references that ground each model in its tradition of methodological development and practical implementation with a focus on connecting researchers to resources outside traditional neuroimaging journals, and (3) provide practical resources in the form of a codebook companion demonstrating how to fit these models. These resources together aim to enhance training for the next generation of developmental neuroscientists by providing a solid foundation for future forays into advanced modeling applications.

Flat portions of the numerical range of a $$6 \times 6$$ companion matrix

Flat portions of the numerical range of a $$6 \times 6$$ companion matrix

Consistency and asymptotic normality in a class of nearly unstable processes

This paper deals with inference in a class of stable but nearly-unstable processes. Autoregressive processes are considered, in which the bridge between stability and instability is expressed by a time-varying companion matrix $$A_{n}$$ with spectral radius $$\rho (A_{n}) < 1$$ satisfying $$\rho (A_{n}) \rightarrow 1$$ . This framework is particularly suitable to understand unit root issues by focusing on the inner boundary of the unit circle. Consistency is established for the empirical covariance and the OLS estimation together with asymptotic normality under appropriate hypotheses when A, the limit of $$A_n$$ , has a real spectrum, and a particular case is deduced when A also contains complex eigenvalues. The asymptotic process is integrated with either one unit root (located at 1 or $$-1$$ ), or even two unit roots located at 1 and $$-1$$ . Finally, a set of simulations illustrate the asymptotic behavior of the OLS. The results are essentially proved by $$L^2$$ computations and the limit theory of triangular arrays of martingales.