THE GENERALIZED REFLEXIVE SOLUTION FOR A CLASS OF MATRIX EQUATIONS （AX- B, XC- D）

In this article, the generalized reflexive solution of matrix equations （AX = B, XC = D） is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.     

MATRIX EQUATION AXB = E WITH PX = sXP CONSTRAINT

The matrix equation AXB = E with the constraint PX=sXP is considered,where P is a given Hermitian matrix satisfying p~2=I and s=±1.By an eigenvalue decomposition of P,the constrained problem can be equivalently transformed to a well-known unconstrained problem of matrix equation whose coefficient matrices contain the corresponding eigenvector, and hence the constrained problem can be solved in terms of the eigenvectors of P.A simple and eigenvector-free formula of the general solutions to the constrained problem by generalized inverses of the coefficient matrices A and B is presented.Moreover,a similar problem of the matrix equation with generalized constraint is discussed.     

Least-Squares Mirrorsymmetric Solution for Matrix Equations （AX=B, XC=D）

In this paper, least-squaxes mirrorsymmetric solution for matrix equations (AX = B, XC = D) and its optimal approximation is considered. With special expression of mirrorsymmetric matrices, a general representation of solution for the least-squares problem is obtained. In addition, the optimal approximate solution and some algorithms to obtain the optimal approximation are provided.     

关于非线性矩阵方程$X^s-A^*X^{-t}A=Q$的Hermitian正定解及其扰动估计

In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X^s - A^＊X^-tA = Q are studied, where Q is a Hermitian positive definite matrix, s and t are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples.     

ON HERMITIAN POSITIVE DEFINITE SOLUTION OF NONLINEAR MATRIX EQUATION X＋A^＊X^-2A=Q

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.     

HGAH矩阵的逆特征值问题

In this paper, the inverse eigenvalue problem of Hermitian generalized anti-Hamihonian matrices and relevant optimal approximate problem are considered. The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived. The solution of the relevant optimal approximate problem is given.     

ON THE RAYLEIGH QUOTIENT FOR SINGULAR VALUES

In this paper, the theoretical analysis for the Rayleigh quotient matrix is studied, some results of the Rayleigh quotient （matrix） of Hermitian matrices are extended to those for arbitrary matrix on one hand. On the other hand, some unitarily invariant norm bounds for singular values are presented for Rayleigh quotient matrices. Our results improve the existing bounds.     

Least-Squares Solution of Inverse Problem for Hermitian Anti-reflexive Matrices and Its Appoximation

In this paper, we first consider the least-squares solution of the matrix inverse problem as follows： Find a hermitian anti-reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X, B we have minA ｜｜AX - B｜｜. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is： Given an arbitrary A^＊, find a matrix A E SE which is nearest to A^＊ in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix.     

矩阵方程ATXA=B的对称正交对称解及其最佳逼近

By applying the generalized singular value decomposition of matrices, this paper provides the necessary and sufficient conditions for the existence and the expression of the symmetric ortho-symmetric solutions of the linear matrix equation A^TXA = B. In addition, the expression of the optimal approximation solution to the given matrix is derived.     

Least-Squares Solutions of the Matrix Equation A^T XA = B Over Bisymmetric Matrices and its Optimal Approximation

A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n 1-j,n 1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A~TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.     

Hermitian自反矩阵反问题的最小二乘解

本文研究了Hermitian自反矩阵反问题的最小二乘解及其最佳逼近.利用矩阵的奇异值分解理论,获得了最小二乘解的表达式.同时对于最小二乘解的解集合,得到了最佳逼近解.     

Filtrations of Totally Reflexive Modules

In this paper, we introduce a subcategory of totally reflexive modules that have a saturated filtration by other totally reflexive modules. We will prove these are precisely the totally reflexive modules with an upper triangular presentation matrix and that such a module has a complete resolution in which every differential can be simultaneously represented by upper triangular matrices.     

Linear restriction problem of Hermitian reflexive matrices and its approximation

In this paper, we consider a linear restriction problem of Hermitian reflexive matrices and its approximation. By using the properties and structure of Hermitian reflexive matrices and the special properties of reflexive vectors and anti-reflexive vectors, we convert the linear restriction problem to an equivalence problem trickily, which is a special feature of this paper and is a different method from other articles. Then we solve this problem completely and also obtain its optimal approximate solution. Moreover, an algorithm provided for it and the numerical examples show that the algorithm is feasible.     

A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.     

线性流形上的广义反射矩阵反问题

设 R∈C^m×m 及 S∈C^n×n 是非平凡Hermitian酉矩阵, 即 R^H=R=R^-1≠±I_m ,S^H=S=S^-1≠±I_n.若矩阵 A∈C^m×n 满足 RAS=A, 则称矩阵 A 为广义反射矩阵.该文考虑线性流形上的广义反射矩阵反问题及相应的最佳逼近问题.给出了反问题解的一般表示, 得到了线性流形上矩阵方程AX₂=Z₂, Y₂^H A=W₂^H 具有广义反射矩阵解的充分必要条件, 导出了最佳逼近问题唯一解的显式表示.     

矩阵方程AXB+CX~TD=E自反最佳逼近解的迭代算法

本文研究了Sylvester矩阵方程AXB+CXTD=E自反(或反自反)最佳逼近解.利用所提出的共轭方向法的迭代算法,获得了一个结果:不论矩阵方程AXB+CXTD=E是否相容,对于任给初始自反(或反自反)矩阵X1,在有限迭代步内,该算法都能够计算出该矩阵方程的自反(或反自反)最佳逼近解.最后,三个数值例子验证了该算法是有效性的.     

四元数矩阵方程(AXB,CXD)=(E,F)的最小范数最小二乘Hermitian解

提出了研究四元数矩阵方程(AXB, CXD)=(E, F)的最小范数最小二乘Hermitian解的一个有效方法.首先应用四元数矩阵的实表示矩阵以及实表示矩阵的特殊结构,把四元数矩阵方程转化为相应的实矩阵方程,然后求出四元数矩阵方程(AXB, CXD)=(E, F)的最小二乘Hermitian解集,进而得到其最小范数最小二乘Hermitian解.所得到的结果只涉及实矩阵,相应的算法只涉及实运算,因此非常有效.最后的两个数值例子也说明了这一点.     

一种求布尔矩阵传递闭包的基于自反矩阵构造的平方算法

首先,介绍布尔矩阵传递闭包的概念及计算问题;随后,分析布尔矩阵的传递闭包和由该布尔矩阵与单位矩阵取并所得到的自反矩阵的传递闭包之间的关系;最后,利用上述结果给出一种求解布尔矩阵传递闭包的基于自反矩阵构造的平方算法,并通过实例说明了其具体计算过程.     

四元数矩阵方程AXAH=B的最小二乘解

引入了四元数矩阵范数的概念，通过使用四无数矩阵的奇异值分解，给出了四元数矩阵方程AXA^H＝B在最小二乘意义下的Hermitian解以及Skew-Hermitian解．