多项式环上一次不定方程的矩阵解法与程序设计

利用多项式环F[x]上的欧几里德算法给出求解多项式环F[x]上的多元一次不定方程通解的矩阵解法，同时利用MATLAB数学软件给出相应的计算机求解多项式环F[x]上一次不定方程的程序，这种方法可以普遍地应用．     

循环线性方程组的求解

利用多项式矩阵理论,给出了循环线性方程组有解的判定并求出各种情况下的解。若方程组有唯一解,求出其唯一解;若方程组有无穷解,求出其极小范数解;若方程组无解,求出其极小范数最小二乘解。     

一类具有非线性互补多项式的奇异矩阵稳定性分析

一类具有非线性互补多项式的奇异矩阵是求解非线性动力学控制系统和模式状态监测的数学基础,分析具有非线性互补多项式的奇异矩阵稳定性,保障控制系统的稳定性。采用共轭梯度法进行奇异分解,提高对具有非线性互补多项式的奇异矩阵双正则函数的边值控制节点的约束能力,结合特征函数在渐进性条件下的Lyapunov-Krasovskii泛函,进行渐进稳定性证明,采用多目标优化局部搜索方法求解奇异矩阵的正则方程组,实现对非线性二阶模糊逻辑系统稳定性控制,求解奇异矩阵的解空间向量,分析其收敛性,根据共轭梯度边值加权优化理论,得到该类具有非线性互补多项式的奇异矩阵的SVD分解具有渐进稳定性的结论。     

随机矩阵特征多项式的相关函数的行列式表示

利用正交多项式的性质给出了高斯辛系综中酉辛群上的随机矩阵特征多项式的相关函数和矩的简洁的行列式表示,且行列式的元为正交多项式。     

行延拓矩阵的矩阵方程组的最佳逼近解

主要讨论行延拓矩阵的线性约束矩阵方程组的最佳逼近;介绍了延拓矩阵的概念;利用矩阵奇异值分解得到了行延拓矩阵的线性约束矩阵方程组有解的充要条件、通解表达式;最后讨论了相应问题的最佳逼近解的表达式。     

分圆多项式在Z[x]中不可约性的新证明

利用对称多项式的基本原理,数论的基本知识及简单的组合公式给出了分圆多项式在有理数域的不可约性的新证明,避免了使用为唯一分解环这一结论。     

分圆多项式在Z[X]中不可约性的新证明

利用对称多项式的基本原理,数论的基本知识及简单的组合公式给出了分圆多项式在有理数域的不可约性的新证明,避免了使用为唯一分解环这一结论。     

矩阵多项式的逆矩阵求解方法

矩阵多项式的知识在很多线性代数教材中的都有所涉及,但是对于矩阵多项式的逆矩阵的计算一般的计算方法比较复杂,本文结合多项式的最大公因式理论与矩阵最小多项式的相关知识,得到了求解一般的矩阵多项式逆矩阵的简单方法。     

用多项式除法求矩阵多项式的逆

以线性代数教学中的一类重要问题"求矩阵多项式的逆矩阵"为例,利用多项式的带余除法,特别是综合除法介绍最常见的一类矩阵多项式求逆的具体计算方法,使得这类问题的求解简单高效、容易掌握。由于这种方法具有一定的构造性,也可使此类问题的解决思路更加清晰,激发学生后续学习的兴趣。     

关于任意体上矩阵方程AXB+CYD=E

求解任意体与环上矩阵方程是近年来矩阵方程研究的一个热点问题，特别是四元数体上矩阵方程的研究。本文利用任意体上的双矩阵分解的方法，讨论了任意体上矩阵方程AXB＋CYD=E，并给出了其一个实用的解法和它的通解表达式。     

Matrix solution bounds of the continuous Lyapunov equation by using a bilinear transformation

By using a bilinear transformation and some linear algebraic techniques, new matrix bounds of the solution of the continuous algebraic Lyapunov equation (CALE) are derived in this paper. Comparing to existing works, these obtained matrix bounds are less restrictive and are easy to be calculated. A numerical example is also given to demonstrate the merits of the present results.     

Symmetric and innerwise matrices for the root-clustering and root-distribution of a polynomial

The general problem of root-clustering and root-distribution of a polynomial in a certain region Γ in the complex plane has been investigated in this paper. The region Γ is general and includes all the previously investigated regions. For the root-clustering problem, it is shown that by using a certain transformation, the necessary and sufficient condition can be represented either in terms of positive definite (p.d.) or positive innerwise (p.i.) matrices. The entries in these matrices are rational functions of the coefficients of the polynomial. The connection between p.d. and p.i. matrices is established in terms of matrix multiplication.     

Solution of linear state-space equations and parameter estimation in feedback systems using laguerre polynomial expansion

This paper analyzes the application of Laguerre polynomial expansion to linear systems. It can be applied to the solution of linear state equations by using an algebraic matrix to determine the coefficients of the Laguerre expansion. It also can be applied to system identification by using the expansion to determine the coefficients in the transfer function. Examples are given to demonstrate the accuracy of finite order expansion by Laguerre polynomials.     

Nonlinear H∞ feedback control with integrator for polynomial discrete-time systems

This paper investigates the problem of designing a nonlinear _H∞$H_{\infty }$ feedback controller for polynomial discrete-time systems with and without polytopic uncertainties. The objective is to design a controller such that the ratio between the energy of the regulated outputs and the energy of the exogenous disturbance/inputs is minimized or guaranteed to be less or equal to a prescribed value. It is well known that the state dependant or parameter dependant Lyapunov function is always chosen for synthesizing polynomial discrete-time systems. This leads the solution to be nonconvex because the Lyapunov function and the controller matrix are coupled and therefore cannot be solved by semidefinite programming (SDP). Hence, in this paper, an integrator is proposed to be incorporated into the controller structure. In doing so, the coupling of Lyapunov function and controller matrix can be eliminated effectively. This somehow simplifies the numerical solution of the problem. Then, by using SOS decomposition approach, sufficient conditions for the existence of the proposed controller are provided in terms of solvability of the state-dependent linear matrix inequalities (SDLMIs) which can be solved by SDP. A tunnel diode circuit is used to demonstrate the effectiveness of this integrator approach.     

A matrix generalization of heaviside's expansion theorem

Starting with the energy and dissipation functions of the general n mesh linear bilateral network and using the operational methods of the Laplacian transformation, a solution is obtained for the Lagrangian equations of the system subject to initial boundary conditions. The equations take a particularly simple and general form if matrix notation is used.It is noted that the general case bears a close resemblance to the simple, one mesh, series circuit when the scalar factors which appear in this circuit are generalized to matrix form.     

The solution to matrix equation AX+XC=B

In this note, the problem of solution to the matrix equation AX+X^TC=B is considered by the Moore-Penrose generalized inverse matrix. A general solution to this equation is obtained. At the same time, some useful conclusions are made, which play important roles in the linear system theories and applications.     

Walsh series approach to lumped and distributed system identification

This paper considers the problem of identifying the parameters of dynamic systems from input-output records. Both lumped-parameter and distributed-parameter systems, deterministic and stochastic, are studied. The approach adopted is that of expanding the system variables in Walsh series. The key point is an operational matrix P which relates the coefficient matrix Г of the Walsh series of a given function with the coefficient matrix of its first derivative. Using this operational matrix P one overcomes the necessity to use differentiated data, a fact that usually is avoided either by integration of the data or by using discrete-time models. Actually, the original differential input-output model is converted to a linear algebraic (or regression) model convenient for a direct (or a least squares) solution. A feature of the method is that it permits the identification of unknown initial conditions simultaneously with the parameter identification. The results are first derived for single-input single-output systems and then are extended to multi-input multi-output systems. The case of non-constant parameters is treated by assuming polynomial forms. Some results are also included concerning the identification of state-space and integral equation models. The theory is supported by two examples, which give an idea of how effective the method is expected to be in the real practice.