正定矩阵的几种经典证明方法

矩阵是数学中一个重要的基本概念,是代数学的一个主要研究对象,同时矩阵论又是研究线性代数的一个有力工具.而正定矩阵因其特有的性质及广泛的应用领域使得很多学者对其进行了大量的研究,本文主要利用特征值,单位矩阵,上三角矩阵,可逆矩阵等知识给出正定矩阵的几种证明方法和一些性质,希望能起到推广正定矩阵应用的作用。     

基于广义逆的桁架结构非线性homologous设计方法

运用M-P广义逆理论,研究了桁架结构的非线性homologous设计问题。将homologous变形约束条件引入结构基本方程,运用M-P广义逆矩阵的性质,将基本方程解的存在条件表示为含可变节点坐标变量的非线性方程组,通过求解该非线性方程组找到了满足homologous变形约束要求的解,并为此推导了AA (A为任意矩阵,A 为A的M-P广义逆矩阵)求偏导数的显式表达。最后的算例验证了本方法的正确性和有效性。     

处理测量数据的方法探究

本文介绍了条件平差中求解平差值的步骤与间接平差求平差值的步骤,在求解过程中都涉及求解法方程系数矩阵的逆,讨论了求解法方程系数矩阵的逆,推倒了二阶、三阶法方程系数矩阵的逆,并结合实例数据进行了应用。     

正互反矩阵的两个定理及其推广

本文在现有对正互反矩阵研究的基础上,进一步深入探讨正互反矩阵的性质:对于任意阶正互反矩阵的所有特征根之和等于该矩阵的阶数;对三阶正互反矩阵的特征根进行了仔细研究并通过盛金公式进行证明,得出三阶正互反矩阵有且只有一个实根和一对共轭虚根。根据研究,对正互反矩阵的性质推广到任意对角线上元素为1的阶矩阵的所有特征根之和进行研究,得出对角线上元素为1的任意阶矩阵,其特征根之和为。     

矩阵特征值求解方法

设A是一个n×n对称矩阵,要解的问题是求出特征值λ和对应的n维向量v,使Av=λv,此问题已有许多方法可解.本文提出一个可对角化的解法,同时对求解向量方程Av=λBv(其中v是向量,B是n×n阵)的特征值和特征向量,提出可化为对称情形的一般特征值问题求解。     

除环上无限方程的解

本文讨论了无限矩阵,得到了除环上无限方程的解存在性定理以及解的结构定理,并进行了举例说明.     

逆H-循环矩阵的相关问题研究

循环矩阵和H-矩阵的研究都是当前矩阵理论和应用研究的热点,根据循环矩阵和H-矩阵的定义,给出了逆H-循环矩阵的定义,并在此基础上推导出了逆H-循环矩阵的一些特殊性质.     

矩阵方程AX+BY+CZ=F广义中心对称最小二乘解

研究矩阵方程AX+BY+CZ=F广义中心对称解,给出了AX+BY+CZ=F的最小二乘广义中心对称解的表达式,导出了AX+BY+CZ=F有广义中心对称解的条件.讨论了在AX+BY+CZ=F最小二乘广义中心对称解集合中求与给定矩阵最佳逼近解.     

复数域上的共轭次辛矩阵

给出了复数域上共轭次辛矩阵的概念,研究了这类矩阵的某些性质,讨论了它的若干判定定理,得出了一些新的结果.     

关于伴随矩阵性质的进一步讨论

进一步讨论了伴随矩阵的秩、可逆性、特征值及一些特殊矩阵的伴随矩阵,并加以证明,力争对伴随矩阵有一个较完整的认识。     

Approximations to the solution of linear Fredholm integrodifferential-difference equation of high order

There are few techniques available to numerically solve linear Fredholm integrodifferential-difference equation of high-order. In this paper we show that the Taylor matrix method is a very effective tool in numerically solving such problems. This method transforms the equation and the given conditions into the matrix equations. By merging these results, a new matrix equation which corresponds to a system of linear algebraic equation is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the differential, difference, differential-difference and Fredholm integral equations. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method.     

Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients

In this study, a practical matrix method is presented to find an approximate solution of high-order linear Fredholm integro-differential equations with constant coefficients under the initial-boundary conditions in terms of Taylor polynomials. The method converts the integro-differential equation to a matrix equation, which corresponds to a system of linear algebraic equations. Error analysis and illustrative examples are included to demonstrate the validity and applicability of the technique.     

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.     

On the matching equations of kinetic energy shaping in IDA-PBC

Interconnection and damping assignment passivity-based control scheme has been used to stabilize many physical systems such as underactuated mechanical systems through total energy shaping. In this method, some partial differential equations (PDEs) related to kinetic and potential energy shaping shall be solved analytically. Finding a suitable desired inertia matrix as the solution of nonlinear PDEs relevant to kinetic energy shaping is a challenging problem. In this paper, a systematic approach to solving this matching equation for systems with one degree of underactuation is proposed. A special structure for desired inertia matrix is proposed to simplify the solution of the corresponding PDE. It is shown that the proposed method is more general than that of some reported methods in the literature. In order to derive a suitable desired inertia matrix, a necessary condition is also derived. The proposed method is applied to three examples, including pendubot, VTOL aircraft, and 2D SpiderCrane.     

基于自标定的两视图三维重构研究

相机标定是三维重构中的关键步骤,本文对在自标定下的两视图三维几何模型恢复进行研究。先采用SURF算法得到图像匹配点对,在此基础上,利用RANSAC鲁棒估计基础矩阵;然后根据Kruppa方程所推导出的二次方程自标定出相机内参数,进一步地,通过分解本质矩阵求解出相机外参数;最后,根据三角化法求得空间三维坐标值。实验结果证实,该方法具有效性和正确性,并且能够真实地再现物体的三维模型。     

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。     

Least squares Hermitian solution of the complex matrix equation AXB+CXD=E with the least norm

In this paper, a new method is presented to solve the least squares Hermitian problem of the complex matrix equation AXB+CXD=E. The explicit expression of least squares Hermitian solution with the least norm is given. The least squares Hermitian solution with the least norm of complex matrix equation AXB=E is also derived. Numerical algorithms and numerical examples show the feasibility of our method.     

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.     

New solution bounds for the continuous algebraic Riccati equation

In this paper, by constructing the equivalent form of the continuous algebraic Riccati equation and utilizing the eigenvalue and singular value inequalities of matrix's sum and product, we propose new lower and upper matrix bounds for the solution of the continuous algebraic Riccati equation. Finally, we give corresponding numerical examples to illustrate the effectiveness of our results.     

New upper bounds on the solution matrix to the continuous algebraic Riccati matrix equation

In this paper, new upper bounds for the solution matrix of the continuous algebraic Riccati matrix equation (CARE) are derived by means of some matrix inequalities and linear algebraic techniques. Furthermore, for the derived each bound, iterative algorithms are developed to obtain sharper solution estimates. Comparing with some appearing results in the literature, the presented bounds are less restrictive and more efficient. Finally, numerical examples are given to illustrate the effectiveness of the proposed results.