On the minimum-norm least squares solution of the complex generalized coupled sylvester matrix equations

By means of the real linear operator, we establish an iterative algorithm for solving a class of complex generalized coupled Sylvester matrix equations. The finite termination of the proposed algorithm is proved. By representing a complex matrix as a larger real matrix, we present a new method to prove that the minimum-norm solution or minimum-norm least squares solution of the complex generalized coupled Sylvester matrix equations can be obtained by an appropriate selection for the initial matrices, which has not been found in the existing work. Numerical experiments on some randomly generated data and practical image restoration problem show that the proposed algorithm is feasible and effective.     

Factor gradient iterative algorithm for solving a class of discrete periodic Sylvester matrix equations

At present, gradient iteration methods have been used to solve various Sylvester matrix equations and proved effective. Based on this method, we generalize the factor gradient iterative method (FGI) for solving forward periodic Sylvester matrix equations (FPSME) and backward periodic Sylvester matrix equations (BPSME). To accelerate the convergence of the iterative method, we refer to Gauss-Seidel and Jacobi iterative construction ideas and use the latest matrix information in the FGI iterative method to obtain the modified factor gradient iterative (MFGI) method. Then, the convergence of the proposed methods and the selection of optimal factors are proved. The last numerical examples illustrate the effectiveness and applicability of the iterative methods.     

An iterative algorithm for generalized periodic multiple coupled Sylvester matrix equations

The paper is indicated to constructing a modified conjugate gradient iterative (MCG) algorithm to solve the generalized periodic multiple coupled Sylvester matrix equations. It can be proved that the proposed approach can find the solution within finite iteration steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm solution of the system. Some numerical examples are illustrated to show the performance of the proposed approach and its superiority over the existing method CG.     

Generalized conjugate direction algorithm for solving generalized coupled Sylvester transpose matrix equations over reflexive or anti-reflexive matrices

The paper studies the iterative solutions of the generalized coupled Sylvester transpose matrix equations over the reflexive (anti-reflexive) matrix group by the generalized conjugate direction algorithm. The convergence analysis shows that the solution group can be obtained within finite iterative steps in the absence of round-off errors for any initial given reflexive (anti-reflexive) matrix group. Furthermore, we can get the minimum-norm solution group by choosing special kinds of initial matrix group. Finally, some numerical examples are given to demonstrate the algorithm considered is quite effective in actual computation.     

Conjugate gradient-based iterative algorithm for solving generalized periodic coupled Sylvester matrix equations

This paper focuses on constructing a conjugate gradient-based (CGB) method to solve the generalized periodic coupled Sylvester matrix equations in complex space. The presented method is developed from a point of conjugate gradient methods. It is proved that the presented method can find the solution of the considered matrix equations within finite iteration steps in the absence of round-off errors by theoretical derivation. Some numerical examples are provided to verify the convergence performance of the presented method, which is superior to some existing numerical algorithms both in iteration steps and computation time.     

Gradient based approach for generalized discrete-time periodic coupled Sylvester matrix equations

The paper is dedicated to solving the generalized periodic discrete-time coupled Sylvester matrix equation, which is frequently encountered in control theory and applied mathematics. The solvable condition and a iterative algorithm for this equation are presented. The proposed method is developed from a point of least squares method. The rationality of the method is testified by theoretical analysis, which shows that the algorithm can solve the problem within finite number of iterations. The presented approach is numerically reliable and requires less computation. A numerical example illustrates the effectiveness of the raised result.     

A finite iterative algorithm for the general discrete-time periodic Sylvester matrix equations

The purpose of this paper is to present an iterative algorithm for solving the general discrete-time periodic Sylvester matrix equations. It is proved by theoretical analysis that this algorithm can get the exact solutions of the periodic Sylvester matrix equations in a finite number of steps in the absence of round-off errors. Furthermore, when the discrete-time periodic Sylvester matrix equations are consistent, we can obtain its unique minimal Frobenius norm solution by choosing appropriate initial periodic matrices. Finally, we use some numerical examples to illustrate the effectiveness of the proposed algorithm.     

Solutions to linear bimatrix equations with applications to pole assignment of complex-valued linear systems

We study in this paper solutions to several kinds of linear bimatrix equations arising from pole assignment and stability analysis of complex-valued linear systems, which have several potential applications in control theory, particularly, can be used to model second-order linear systems in a very dense manner. These linear bimatrix equations include generalized Sylvester bimatrix equations, Sylvester bimatrix equations, Stein bimatrix equations, and Lyapunov bimatrix equations. Complete and explicit solutions are provided in terms of the bimatrices that are coefficients of the equations/systems. The obtained solutions are then used to solve the full state feedback pole assignment problem for complex-valued linear system. For a special case of complex-valued linear systems, the so-called antilinear system, the solutions are also used to solve the so-called anti-preserving (the closed-loop system is still an antilinear system) and normalization (the closed-loop system is a normal linear system) problems. Second-order linear systems, particularly, the spacecraft rendezvous control system, are used to demonstrate the obtained theoretical results.     

Symmetric least squares solution of a class of Sylvester matrix equations via MINIRES algorithm

The purpose of this paper is deriving the minimal residual (MINIRES) algorithm for finding the symmetric least squares solution on a class of Sylvester matrix equations. We prove that if the system is inconsistent, the symmetric least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrix to obtain the minimum norm least squares symmetric solution of the problem. Finally, we give some numerical examples to illustrate the performance of MINIRES algorithm.     

Widely linear estimation for multisensor quaternion systems with mixed uncertainties in the observations

The optimal widely linear state estimation problem for quaternion systems with multiple sensors and mixed uncertainties in the observations is solved in a unified framework. For that, we devise a unified model to describe the mixed uncertainties of sensor delays, packet dropouts and uncertain observations by using three Bernoulli distributed quaternion random processes. The proposed model is valid for linear discrete-time quaternion stochastic systems measured by multiple sensors and it allows us to provide filtering, prediction and smoothing algorithms for estimating the quaternion state through a widely linear processing. Simulation results are employed to show the superior performance of such algorithms in comparison to standard widely linear methods when mixed uncertainties are present in the observations.     

Compressed sensing for real measurements of quaternion signals

The paper concerns compressed sensing methods in the quaternion algebra. We prove that it is possible to uniquely reconstruct – by ?₁ norm minimization – a sparse quaternion signal from a limited number of its real linear measurements, provided the measurement matrix satisfies so-called restricted isometry property with a sufficiently small constant. We also provide error estimates for the approximated reconstruction of a non-sparse quaternion signal from noisy and noiseless data.     

射影空间上的等参子流形

讨论了复射影空间ＣＰｎ和四元数射影空间ＨＰｎ上的等参子流形．利用ＣＰｎ上等参子流形的分类定理［１０］,证明了ＣＰｎ上等参子流形的刚性定理．同时,还得到了四元数射影空间ＨＰｎ上等参子流形的例子．     

Class-aware tensor factorization for multi-relational classification

In this paper, we propose a tensor factorization method, called CLASS-RESCAL, which associates the class labels of data samples with their latent representations. Specifically, we extend RESCAL to produce a semi-supervised factorization method that combines a classification error term with the standard factor optimization process. CLASS-RESCAL assimilates information from all the relations of the tensor, while also taking into account classification performance. This procedure forces the data samples within the same class to have similar latent representations. Experimental results on several real-world social network data indicate this is a promising approach for multi-relational classification tasks.     

Cyclic codes over some finite quaternion integer rings

In this paper, cyclic codes are studied over some finite quaternion integer rings with respect to the quaternion Mannheim distance.     

A relaxed MSIO iteration algorithm for solving coupled discrete Markovian jump Lyapunov equations

In this paper, combining the multi-step Smith-inner-outer (MSIO) iteration framework with some tunable parameters, a relaxed MSIO iteration method is proposed for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) in the discrete-time jump linear systems with Markovian transitions. The convergence properties of the relaxed MSIO iteration method are investigated, and the choices of the parameters are also discussed. In order to accelerate the convergence rate of the relaxed MSIO iteration method for solving the CLMEs, a current-estimation-based and a weighted relaxed MSIO iteration algorithms are presented, respectively. Finally, several numerical examples are given to verify the superiorities of the proposed relaxed algorithms.     

A modified noise-tolerant ZNN model for solving time-varying Sylvester equation with its application to robot manipulator

Recently, Xiao et al. (2021) proposed an efficient noise-tolerant zeroing neural network (NTZNN) model with fixed-time convergence for solving the time-varying Sylvester equation. In this paper, we propose a modified version of their NTZNN model, named the modified noise-tolerant zeroing neural network (MNTZNN) model. It extends the NTZNN model to a more general form and then we prove that, with appropriate parameter selection, our new MNTZNN model can significantly accelerate the convergence of the NTZNN model. Numerical experiments confirm that the MNTZNN model not only maintains fixed-time convergence and noise-tolerance but also has a faster convergence rate than the NTZNN model under certain conditions. In addition, the design strategy of the MNTZNN is also successfully applied to the path tracking of a 6-link planar robot manipulator under noise disturbance, which demonstrates its applicability and practicality.     

四元数分形奇异值分解方法与彩色图像重建

基于四元数矩阵彩色图像奇异值分解,得到表征彩色图像的不同分量的奇异值,运用分形快速确定图像的拐点。该方法具有快速和简单可行的优点。以受噪声污染图像为例,该方法针对彩色图像去噪具有较好的效果。     

二阶系统数值解耦方法的研究

数值代数领域通过保持Lancaster结构来研究二阶系统的解耦问题,但寻找解耦变换涉及到了非线性方程组求解问题,难以实现. 提出了一种二阶系统数值解耦的新方法. 根据系统解耦前后的同谱信息确定解耦后的系统,将寻找解耦变换的非线性问题转化为齐次Sylvester方程求解问题; 并利用矩阵的Kronecker积理论求解二阶系统的解耦变换. 数值试验证明了该方法的可行性,为二阶系统的数值解耦找到了更便易的实现途径.     

Experimental measurement of the quantum geometric tensor using coupled qubits in diamond

Geometry and topology are fundamental concepts, which underlie a wide range of fascinating physical phenomena such as topological states of matter and topological defects. In quantum mechanics, the geometry of quantum states is fully captured by the quantum geometric tensor. Using a qubit formed by an NV center in diamond, we perform the first experimental measurement of the complete quantum geometric tensor. Our approach builds on a strong connection between coherent Rabi oscillations upon parametric modulations and the quantum geometry of the underlying states. We then apply our method to a system of two interacting qubits, by exploiting the coupling between the NV center spin and a neighboring ¹³C nuclear spin. Our results establish coherent dynamical responses as a versatile probe for quantum geometry, and they pave the way for the detection of novel topological phenomena in solid state.     

A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations

In this paper, we discuss the properties of the eigenvalues related to the symmetric positive definite matrices. Several new results are established to express the structures and bounds of the eigenvalues. Using these results, a family of iterative algorithms are presented for the matrix equation AX=F and the coupled Sylvester matrix equations. The analysis shows that the iterative solutions given by the least squares based iterative algorithms converge to their true values for any initial conditions. The effectiveness of the proposed iterative algorithm is illustrated by a numerical example.