Matrix Inequalities Research Articles

An effective Hamiltonian for the simulation of open quantum molecular systems

We discuss the derivation of an effective Hamiltonian for open quantum many-particle systems. The aim is to define an operator that can be used for (molecular) simulations where, through the exchange of energy and matter with the surrounding environment (reservoir), the number of particles, n, becomes a variable of the problem. The Hamiltonian is formally derived from the Von Neumann equation; specifically, we derive an n-hierarchy of equations for the density matrix, ρ^n , for near equilibrium situations. Such a hierarchy, in case of stationary equilibrium, delivers the standard grand canonical density matrix as it would be expected. We report that a similar Hamiltonian was conjectured, from empirical considerations, in the field of superconductivity. Thus our result also provide a formal basis for this long-standing hypothesis. Finally, an application is discussed for Path Integral simulations of molecular systems.

Quaternion Differential Matrix Equations with Singular Coefficient Matrices

Quaternion Differential Matrix Equations with Singular Coefficient Matrices

Modeling of Mean-Value-at-Risk Investment Portfolio Optimization Considering Liabilities and Risk-Free Assets

This paper aims to design a quadratic optimization model of an investment portfolio based on value-at-risk (VaR) by entering risk-free assets and company liabilities. The designed model develops Markowitz’s investment portfolio optimization model with risk aversion. Model development was carried out using vector and matrix equations. The entry of risk-free assets and liabilities is essential. Risk-free assets reduce the loss risk, while liabilities accommodate a fundamental analysis of the company’s condition. The model can be applied in various sectors of capital markets worldwide. This study applied the model to Indonesia’s mining and energy sector. The application results show that risk aversion negatively correlates with the mean and VaR of the return of investment portfolios. Assuming that risk aversion is in the 5.1% to 8.2% interval, the maximum mean and VaR obtained for the next month are 0.0103316 and 0.0138270, respectively, while the minimum mean and VaR are 0.0102964 and 0.0137975, respectively. The finding of this study is that the vector equation for investment portfolio weights is obtained, which can facilitate calculating investment portfolio weight optimization. This study is expected to help investors control the quality of appropriate investment, especially in some stocks in Indonesia’s mining and energy sector.

Data-driven adaptive optimal control for discrete-time linear time-invariant systems

In this paper, the problem of data-driven quadratic optimal control is investigated for discrete-time linear systems without the exact knowledge of system dynamics. A value iteration-based algorithm is proposed to obtain the optimal feedback gain. The main idea of the proposed approach is to obtain two sequences for the approximation of the unique positive definite solution of the corresponding algebraic Riccati matrix equation and the optimal feedback gain by using the collected data of the system states and control inputs. The proposed algorithm could be activated by an arbitrary bounded control input and a simple initial value of the Lyapunov matrix. An important feature of this algorithm is that an original stabilising feedback gain is not required. Finally, the effectiveness of the proposed approach is verified by two examples.

An accelerated zeroing neural network for solving continuous coupled Lyapunov matrix equations

AbstractIn this paper, an improved zeroing neural network (ZNN) model is proposed to obtain the positive definite solutions of the continuous coupled Lyapunov matrix equations (CLMEs) associated with continuous‐time Markovian jump (CMJ) systems. To achieve this, a general ZNN model is established by constructing a matrix‐valued error function. Then, to accelerate the convergence rate of the proposed ZNN model, the latest estimation is introduced to obtain an improved ZNN model. Some convergence conditions have been derived for the presented improved ZNN model through Lyapunov theory. Comparisons among the improved ZNN model and the existing results are conducted to illustrate the advantages of the proposed improved ZNN model in numerical examples.

Massless spin-2 field solutions with spherical symmetry: Eliminating gauge degrees of freedom

We examine the basic matrix equation for a spin-2 field in spherical coordinates and a tetrad of Minkowski spacetime. The technique of Wigner D-functions is used to separate the variables. The operators of energy, the square and the third projection of the total angular momentum, and the space reflection operators are then diagonalized on the solutions. The separation of the variables is done, the problem reduces to 11 radial second-order differential equations for 11 variables related to the scalar and symmetric tensors. We then consider the case of a massless field. The four gauge solutions for the massless spin-2 field have been found in explicit form. These gauge solutions do not contribute to physically observable quantities such as the energy–momentum tensor. They are derived according to Pauli–Fierz theory from four independent spherical solutions for a spin-1 massless particle. We have also constructed two types of solutions, related to opposite parities, which are not the gauge solutions. These solutions describe physically observable states with spherical symmetry for the spin-2 massless field.

Orthogonality and unitary conditions for solutions to some consistent linear matrix equations

The nonlinear matrix equation (A1 + B1X1D1)(A2 + B2X2D2) = A, presents a matrix identity, where A, Ai, Bi, and Di are known matrices of suitable sizes and Xi are unknown matrices for i = 1,2, over the field of complex numbers ℂ. Several known simple methods in linear algebra can handle the orthogonality problem. In this paper, we select some linear matrix equations as illustrative examples to discuss some properties of pairs of identically dimensional consistent linear matrix equations, then apply specific matrix analytic tools, among them is the matrix rank method, the rank of a matrix is one of the most basic quantities and useful methods and tools that are widely used in linear algebra specifically, in matrix theory and its applications. We consider certain forms of linear matrix equations such as T1X1 = N1 and the system A3X3 = C3, X3B3 = D3, to derive novel conditions dictating the orthogonality of all solutions to two consistent matrix equations, as well as the necessary and sufficient conditions for all solutions to be unitary, further we give an illustrative example.

The relaxed gradient based iterative algorithm for solving the generalized coupled complex conjugate and transpose Sylvester matrix equations

Inspired by the idea of Ma et al. (Journal of the Franklin Institute, 2018), we adopt relaxation technique and introduce relaxation factors into the gradient based iterative (GI) algorithm, and the relaxed based iterative (RGI) algorithm is established to solve the generalized coupled complex conjugate and transpose Sylvester matrix equations. By applying the real representation and straighten operation, we contain the sufficient and necessary condition for convergence of the RGI method. In order to effectively utilize this algorithm, we further derive the optimal convergence parameter and some related conclusions. Moreover, to overcome the high dimension calculation problem, a sufficient condition for convergence with less computational complexity is determined. Finally, numerical examples are reported to demonstrate the availability and superiority of the constructed iterative algorithm.

Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse

We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting an m×m positive definite matrix. The DOIA is a fast, convergent, iterative algorithm. Some properties and the estimation of residual error of the DOIA are given to prove the absolute convergence. Numerical tests demonstrate the usefulness of the double-optimal solution (DOS) and DOIA in solving square or nonsquare linear matrix equations and in inverting nonsingular square matrices. To speed up the convergence, a restarted technique with frequency m is proposed, namely, DOIA(m); it outperforms the DOIA. The pseudoinverse of a rectangular matrix can be sought using the DOIA and DOIA(m). The Moore–Penrose iterative algorithm (MPIA) and MPIA(m) based on the polynomial-type matrix pencil and the optimized hyperpower iterative algorithm OHPIA(m) are developed. They are efficient and accurate iterative methods for finding the pseudoinverse, especially the MPIA(m) and OHPIA(m).

Unified method for typical gear failure modeling and stiffness calculation based on the matrix equation

Unified method for typical gear failure modeling and stiffness calculation based on the matrix equation

A Dynamic Parameter Noise-Tolerant Zeroing Neural Network for Time-Varying Quaternion Matrix Equation With Applications.

As a common and significant problem in the field of industrial information, the time-varying quaternion matrix equation (TV-QME) is considered in this article and addressed by an improved zeroing neural network (ZNN) method based on the real representation of the quaternion. In the light of an improved dynamic parameter (IDP) and an innovative activation function (IAF), a dynamic parameter noise-tolerant ZNN (DPNTZNN) model is put forward for solving the TV-QME. The presented IDP with the character of changing with the residual error and the proposed IAF with the remarkable performance can strongly enhance the convergence and robustness of the DPNTZNN model. Therefore, the DPNTZNN model possesses fast predefined-time convergence and superior robustness under different noise environments, which are theoretically analyzed in detail. Besides, the provided simulative experiments verify the advantages of the DPNTZNN model for solving the TV-QME, especially compared with other ZNN models. Finally, the DPNTZNN model is applied to image restoration, which further illustrates the practicality of the DPNTZNN model.

Solution to Several Split Quaternion Matrix Equations

Split quaternions have various applications in mathematics, computer graphics, robotics, physics, and so on. In this paper, two useful, real representations of a split quaternion matrix are proposed. Based on this, we derive their fundamental properties. Then, via the real representation method, we obtain the necessary and sufficient conditions for the existence of solutions to two split quaternion matrix equations. In addition, two experimental examples are provided to show their feasibility.

The (anti‐)η$$ \eta $$‐Hermitian solution to a novel system of matrix equations over the split quaternion algebra

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti‐) ‐Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti‐) ‐Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.

Aerodynamic modeling and performance analysis of model predictive controller for fixed wing vertical takeoff and landing unmanned aerial vehicle

This research focuses mainly on the aerodynamic modelling and performance analysis of a model predictive controller for a hybrid fixed-wing vertical takeoff and landing of unmanned aerial vehicle. The aerodynamics, which comprises several aerodynamic characteristics including the lift, drag, and thrust coefficients, is modelled using Newton’s second law of motion. The force and moment equations were obtained, and they were then converted into matrix and state equation form, together with the transformation matrices. The essential equations with six degrees of freedom (6 DoF) and 12 state matrices including the control input and manipulating variables were obtained. The proposed model predictive controller (MPC) was designed using the optimal model predictive controller design parameters, such as a sampling time of 0.1, a prediction horizon of 15, a control horizon of 3 and additional controller settings. With a settling period of 3 s, an overshoot of 0.5865, and a steady state inaccuracy of 0.00785, the MATLAB simulation demonstrates that the system variables, including roll, pitch, and yaw, are stabilized. This MPC control is more effective in anticipating and optimizing the UAV than other control strategies. Eventually, the controller’s simulation on MATLAB Simulink demonstrates the controller’s ability to stabilize and control the system in a real-time application. Autonomous vertical takeoff and landing operations depend heavily on the mathematical model and architecture of the flight controller. Among the uses are inspection, monitoring, and rescue.

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

Ribbed elastic metasurface with lateral scalability for flexural wave manipulation

With the goal of engineering applications, the scalability and structural portability of elastic metasurfaces have become significant and attractive. However, most existing elastic metasurfaces cannot be arbitrarily scalable, which will inevitably limit their applications. Here, a novel conceptual design of ribbed elastic metasurfaces (REMs) is proposed for anomalous wavefront manipulation of flexural waves in thin plates. Ribbed subunits, as functional subunits of REMs, are characterized by simple constructions. The phase shift mechanism for the flexural waves across the ribbed subunits is revealed by analyzing the lowest dispersion bands. The analytical model for ribbed subunits is also established based on the equivalent orthotropic plate theory and transfer matrix method (TMM) to accurately predict the phase shift and amplitude of the transmitted waves. In addition, lateral scalability can be observed when the ratio of rib width to subunit width is constant, owing to the constant equivalent bending stiffness. By considering the lateral scalability of subunits, the continuous adjustment of the refraction angle can be perfectly achieved. The deflecting and focusing functionalities of the designed REMs are numerically and experimentally demonstrated. Both simulated and experimental results are consistent with theoretical results. Since ribbed plates are typical load-bearing structures in engineering, our design has broad application prospects, including but not limited to vibration control, energy harvesting, and structural health monitoring.

Tests for Large-Dimensional Shape Matrices via Tyler’s M Estimators

Tyler’s M estimator, as a robust alternative to the sample covariance matrix, has been widely applied in robust statistics. However, classical theory on Tyler’s M estimator is mainly developed in the low-dimensional regime for elliptical populations. It remains largely unknown when the parameter of dimension p grows proportionally to the sample size n for general populations. By using the eigenvalues of Tyler’s M estimator, this article develops tests for the identity and equality of shape matrices in a large-dimensional framework where the dimension-to-sample size ratio p/n has a limit in (0, 1). The proposed tests can be applied to a broad class of multivariate distributions including the family of elliptical distributions (see model (2.1) for details). To analyze both the null and alternative distributions of the proposed tests, we provide a unified theory on the spectrum of a large-dimensional Tyler’s M estimator when the underlying population is general. Simulation results demonstrate good performance and robustness of our tests. An empirical analysis of the Fama-French 49 industrial portfolios is carried out to demonstrate the shape of the portfolios varying. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

A Low-Rank Global Krylov Squared Smith Method for Solving Large-Scale Stein Matrix Equation

A Low-Rank Global Krylov Squared Smith Method for Solving Large-Scale Stein Matrix Equation

Programmable Wave‐Based Meta‐Computer

AbstractIn light of the escalating demand for computing powers, wave‐based analog computers have emerged as a compelling study area due to their remarkable capacity for ultra‐fast operation. Here, a programmable wave‐based computer, based on programmable meta‐structures is proposed, which is also called as meta‐computer. The meta‐computer can perform various types of analog computations in real‐time in the electromagnetic (EM) wave space, including matrix‐vector multiplication, discrete Fourier transform, filtering, and solving complex matrix equations. Owing to the powerful capability of the programmable meta‐structures in manipulating the EM fields and waves in real‐time, the proposed meta‐computer exhibits the advantages of flexible re‐programmability, low complexity, and direct computation in the EM space. Two prototypes of the meta‐computer in the microwave frequency are designed and fabricated, and the computing capability is validated through both numerical simulations and experiments. This work contributes a prominent approach for fully programmable wave‐based computers with noteworthy implications for expanding the applications of analog computing.

Optimization of some types of Rényi divergences between unitary orbits

In this article, we investigate explicitly the minimal and maximal values of the α-z-Rényi divergences and some other types of Rényi divergences between unitary orbits. Our main tools are the majorization theory of eigenvalues of matrices, and some matrix inequalities such as Golden–Thompson's and Araki–Lieb–Thirring's inequalities.