Minor Self-conjugate and Skewpositive Semidefinite Solutions to a System of Matrix Equations over Skew Fields

Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and sufficient conditions for the existence of and the expressions for the msc solutions and the ssd solutions are obtained for the system.

THE SYMMETRIC AND SYMMETRIC POSITIVE SEMIDEFINITE SOLUTIONS OF LINEAR MATRIX EQUATION——B^TXB = D ON LINEAR MANIFOLDS

This paper discusses the solutions of the linear matrix equation BT X B=Don some linear manifolds.Some necessary and sufficient conditions for the existenceof the solution and the expression of the general solution are given.And also someoptimal approximation solutions are discussed.

COMPUTING THE NEAREST BISYMMETRIC POSITIVE SEMIDEFINITE MATRIX UNDER THE SPECTRAL RESTRICTION

Let A and C denote real n × n matrices. Given real n-vectors x1,……… ,xm,m≤n,and a set of numbers (L)={λ1,λ2…λm}，We descrbe(Ⅰ)the set( ) of all real n × n bisymmetric positive seidefinite matrices A such that Axi is the "best"approximate to λixi, i = 1, 2,..., m in Frobenius norm and (Ⅱ) the Y in set ( )which minimize Frobenius norm of ||C - Y||.An existence theorem of the solutions for Problem Ⅰ and Problem Ⅱ is given andthe general expression of solutions for Problem Ⅰ is derived. Some sufficient conditionsunder which Problem Ⅰ and Problem Ⅱ have an explicit solution is provided. A numer-ical algorithm of the solution for Problem Ⅱ has been presented.

Generalized bipositive semidefinite solutions to a system of matrix equations

In this paper, a system of complex matrix equations was studied. Necessary and sufficient conditions for the existence and the expression of generalized bipositive semidefinite solution to the system were given. In addition, a criterion for a matrix to be generalized bipositive semidefinite was determined.

THE POSITIVE SEMIDEFINITE SOLUTION OF THE MATRIX EQUATION (A^TXA, B^TXB) = (C, D)

In this paper, we consider the positive semidefinite solution of the matrix equation (AT X A, BT X B) - (C, D). A necessary and sufficient condition for the existence of such solution is derived using the generalized singular value decomposition.The general forms of positive semidefinite solution are given.

THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE SOLVABILITY OF A CLASS OF THE MATRIX INVERSE PROBLEM

Censider the solutions of the matrix inverse problem, which are symmetric positive semide finite on a subspace. Necessary and sufficient conditions for the solvability, as well as the general solution are obtained. The best approximate solution by the above solution set is given. Thus the open problem in [1] is solved.

A MATRIX EQUATION FROM AN INVERSE PROBLEM OF VIBRATION THEORY

The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.

几个酉不变范数不等式

利用函数的凸性,得到矩阵酉不变范数的几个不等式.所得不等式改进了一些已有的结果.

LEAST-SQUARES SOLUTIONS OF X^TAX = B OVER POSITIVE SEMIDEFINITE MATRIXES A

This paper is mainly concerned with solving the following two problems: Problem Ⅰ. Given X ∈ Rn×m, B . Rm×m. Find A ∈ Pn such thatwhereProblem Ⅱ. Given A ∈Rn×n. Find A ∈ SE such thatwhere F is Frobenius norm, and SE denotes the solution set of Problem I.The general solution of Problem I has been given. It is proved that there exists a unique solution for Problem II. The expression of this solution for corresponding Problem II for some special case will be derived.

Half thresholding eigenvalue algorithm for semidefinite matrix completion

The semidefinite matrix completion(SMC) problem is to recover a low-rank positive semidefinite matrix from a small subset of its entries. It is well known but NP-hard in general. We first show that under some cases, SMC problem and S1/2relaxation model share a unique solution. Then we prove that the global optimal solutions of S1/2regularization model are fixed points of a symmetric matrix half thresholding operator. We give an iterative scheme for solving S1/2regularization model and state convergence analysis of the iterative sequence.Through the optimal regularization parameter setting together with truncation techniques, we develop an HTE algorithm for S1/2regularization model, and numerical experiments confirm the efficiency and robustness of the proposed algorithm.

半正定的中心对称矩阵反问题

讨论了一类半正定的中心对称矩阵反问题 ,得到了解的具体表达式 ;并就这类矩阵的最佳逼近问题进行了讨论 ,得到了解的存在唯一性 .

矩阵的保半正定性

利用经典的半正定Hermite矩阵的等价条件,讨论了2×2分块矩阵的保半正定性问题.A为2×2半正定Hermite分块矩阵时,则对每一子块分别取迹、行列式、谱范数、秩、数值域后所成矩阵仍为半正定;当A为2×2分块矩阵时,的范数和数值域半径分别不超过的范数和数值域半径.

线性矩阵不等式及在控制理论中的应用

系统和控制理论中许多重要的问题 ,都可转化为线性矩阵不等式约束的凸优化问题 ,从而使其在数值上易于求解。本文阐述了线性矩阵不等式方法的基本概念和内容 ,并介绍了有关算法及计算软件 。

线性流行上一类矩阵方程的对称和对称半正定最小二乘解

导出了在两类线性流形上对称矩阵类和对称半正定矩阵类中一类矩阵方程的最小二乘解的一般表达式,并讨论了解对于已知矩阵的最佳逼近问题.

关于几何凸函数的不等式

首次给出几何凸函数的概念 ,并获得了几何凸函数的若干重要不等式 .

复正规亚半正定矩阵

研究复正规亚半正定矩阵 。

矩阵乘积的特征值与奇异值不等式

获得矩阵乘积的特征值与奇异值的不等式。(1 )设 A、B为非负定 Hermite阵 ,1≤ i1 <… <ik≤ l≤ n,则∏kt=1λl- it+1 (AB)≥ ∏kt=1λl- t+1 (A)λn- it+1 (B) ;∏kt=1λn- l+it(AB)≤ ∏kt=1λn- l+t(A)λit(B)。 (2 )设 A、B∈ Cn× n,1≤ i1 <… <ik≤ l≤ n,则∏kt=1σl- it+1 (AB)≥ ∏kt=1σl- t+1 (A)σn- it+1 (B) ;∏kt=1σn- l+it(AB)≤ ∏kt=1σn- l+t(A)σit(B)。

酉不变范数不等式

针对酉不变范数不等式问题,先给出一组新的标量不等式,进而利用新标量不等式与谱分解定理,得到关于矩阵酉不变范数新的不等式。与前人的结果相比较,新不等式推广了相关文献的结果。

逆向Styan矩阵不等式的推广

对Hermitian半正定矩阵来说,无论其经典形式还是推广形式,Styan矩阵不等式都是互为确定的,因此可称为互逆矩阵不等式.本文给出了一对互逆的无约束条件的Hermitian半正定矩阵的矩阵不等式,以此为基本工具,得到了原已有文献给出的Hermitian半正定矩阵的Styan矩阵不等式的逆向表达式.由文中讨论方法可得到这些互逆的矩阵不等式等式成立的充分必要条件.

关于极值四元数矩阵

得到了关于半正定四元数矩阵迹的一个等式成立的充要条件．