体上的矩阵方程AXA=B

设Ｆ和Ω分别表示一个对合反自同构的体，一个加强Ｐ除环，本文定义了Ω上的亚（半）正定矩阵，给出了矩阵方程ＡＸＡ＾＊＝Ｂ在Ｆ上有（斜）自共轭矩阵解及在Ω上有亚（半）正定矩阵解的充要条件及其解集的显式表示。     

关于矩阵张量积的一类问题

本文给出有限个矩阵张量积分别是正规矩阵、厄米特矩阵、正定矩阵的条件．推广了Y．E．Kuo的相关结果．另外也给出了两个亚半正定矩阵的张量积还是亚半正定矩阵的充要条件．     

一类亚半正定矩阵的左右逆特征值问题

1．引言在工程技术中常常遇到这样一类逆特征值问题：要求在一个矩阵集合S中，找具有给定的部分右特征对（特征值及相应的特征向量）和给定的部分左特征对（特征值及相应的特征向量）的矩阵．文［2］，［3］讨论了S为。x。实矩阵集合的情形．文［4］－［7］对S为nxn实对称矩阵．对称正定矩阵，对称半正定矩阵集合的情形进行了讨论．文【川讨论了S为亚正定阵集合的情形．并提到了对于亚半正定矩阵的情形目下无人涉及，有待进一步研究．本文将对S为nxn亚半正定矩阵集合的情形进行讨论．给出了亚半正定矩阵的左右逆特征值问题有解的充要条件…     

一类亚正定矩阵上的逆向Hadamard不等式和逆向Szasz不等式

利用亚正定矩阵的基本理论,建立了一类亚正定矩阵上的逆向Hadamard不等式和逆向Szasz不等式.     

亚正定矩阵的Bergstrom型不等式

首先证明亚正定矩阵的一个偏序,利用该偏序得到了亚正定矩阵的一些Bergstrom型不等式,推广了近期关于亚正定矩阵行列式不等式的一些结果.     

体上一矩阵方程组的次自共轭及斜亚半正定解

给出了体上的矩阵方程组[AmnXmn=AsnXnm=Osn]有次自共轭解和斜亚半正定解的充要条件及其通解表达式。     

复矩阵的亚半正定性

复亚半正定矩阵是 Hermite正定阵的推广 ,研究了它的 Kronecker积、Hadamard积和行列式理论 ,将实对称阵的 Schur定理、华罗庚定理、Minkowski不等式、Ky-Fan不等式、Ostrowski-Taussky不等式推广到了一类非 Hermite复矩阵上 ,扩大了 Minkowski不等式的指数范围 ,削弱了华罗庚不等式的条件 .     

次亚正定矩阵的次L(o)wner偏序

本文研究了次亚正定矩阵子阵的次L(o)wner偏序,利用次Lwner偏序,获得了几个用低阶矩阵的次亚正定性判别高阶矩阵次亚正定性的充要条件.     

矩阵方程AXAT+BYBT=C的亚半正定解

讨论了矩阵方程AXAT+BYBT=C关于亚半正定矩阵X,Y有解的充要条件,并在有解时给出了解的通式.     

三对角矩阵的亚正定性

讨论了比三对角矩阵更广泛的一类矩阵的亚正定性,从而给出了三对角矩阵是亚正定矩阵的充分条件.     

四元数体上矩阵方程的自共轭解

利用矩阵的M-P逆和矩阵分块,给出了四元数体上矩阵方程XB=D在子空间上有自共轭解的充要条件以及解的一般形式,并由此给出了矩阵方程AXB=D有自共轭解的充要条件和解的一般形式.     

一类矩阵方程的公共解

By applying the GSVD of matrix pairs,we discuss common solutions of the matrix equations AXC = E, BXD = F, AXD = G, BXC = H, under consistent and nonconsistent case respectively. We also discuss common symmetric solutions of the matrix equations AXA^T = E, BXB^T = F, AXB^T = G, BXA^T = H under consistent and nonconsistent case respectively. The necessary and sufficient conditions for the existence and the expressions of solutions of these matrix equations are provided.     

Matrix correction of a dual pair of improper linear programming problems

A matrix is sought that solves a given dual pair of systems of linear algebraic equations. Necessary and sufficient conditions for the existence of solutions to this problem are obtained, and the form of the solutions is found. The form of the solution with the minimal Euclidean norm is indicated. Conditions for this solution to be a rank one matrix are examined. On the basis of these results, an analysis is performed for the following two problems: modifying the coefficient matrix for a dual pair of linear programs (which can be improper) to ensure the existence of given solutions for these programs, and modifying the coefficient matrix for a dual pair of improper linear programs to minimize its Euclidean norm. Necessary and sufficient conditions for the solvability of the first problem are given, and the form of its solutions is described. For the second problem, a method for the reduction to a nonlinear constrained minimization problem is indicated, necessary conditions for the existence of solutions are found, and the form of solutions is described. Numerical results are presented.     

Solving Boundary Value Problems for the Matrix Equation $X^{(2)}(t)-AX(t)=F(t)$

In this paper we present a method for solving the matrix differential equation $X^{(2)}(t)-AX(t)=F(t)$, without increasing the dimension of the problem. By introducing the concept of co-square root of a matrix, existence and uniqueness conditions for solutions of boundary value problems related to the equation as well as explicit solutions of these solutions are given, even for the case where the matrix $A$ has no square roots.     

All Solutions of the Yang-Baxter-like Matrix Equation When $A^3 = A$

Let $A$ be a square matrix satisfying $A^3 = A$. We find all solutions of the Yang-Baxter matrix equation $AXA=XAX$, based on our previous result on all the solutions of the same equation for a matrix $A$ such that $A^2 = I$.     

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger equation

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one-soliton solutions and two-soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.     

一类矩阵方程的埃尔米特自反最小二乘解

利用埃尔米特自反矩阵的表示定理和矩阵的拉直方法,研究了矩阵方程$AX+BY=C$的埃尔米特自反最小二乘问题,进一步,给出了方程在埃尔米特自反矩阵集合中可解的充分必要条件,得到解的一般表达式,最后,对任意给定的一对复矩阵,得到了其相关最佳逼近问题解的表达式.     

一类矩阵方程的双对称定秩解及其最佳逼近

本文研究了矩阵方程AX=B的双对称最大秩和最小秩解问题.利用矩阵秩的方法,获得了矩阵方程AX=B有最大秩和最小秩解的充分必要条件以及解的表达式,同时对于最小秩解的解集合,得到了最佳逼近解.     

Vector least-squares solutions for coupled singular matrix equations

The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian's work and Lev-Ari's results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case.     

Darboux transformations for a matrix long-wave–short-wave equation and higher-order rational rogue-wave solutions

A new matrix long-wave–short-wave equation is proposed with the of help of the zero-curvature equation. Based on the gauge transformation between Lax pairs, both onefold and multifold classical Darboux transformations are constructed for the matrix long-wave–short-wave equation. Resorting to the classical Darboux transformation, a multifold generalized Darboux transformation of the matrix long-wave–short-wave equation is derived by utilizing the limit technique, from which rogue wave solutions, in particular, can be obtained by employing the generalized Darboux transformation. As applications, we obtain rogue-wave solutions of the long-wave–short-wave equation and some explicit solutions of the three-component long-wave–short-wave model, including soliton solutions, breather solutions, the first-order and higher-order rogue-wave solutions, and others by using the generalized Darboux transformation.