Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C

This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation Xm+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

SYMMETRIC POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATIONS (AX,XB)=(C,D) AND AXB=C

The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.

Some new bound estimates of the Hermitian positive definite solutions of the nonlinear matrix equation X^s+ A~*X^(-t) A = Q

The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.

THE OPPENHEIM-TYPE INEQUALITIES FORTHE HADAMARD PRODUCT OF M-MATRIXAND POSITIVE DEFINITE MATRIX

For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).

A Matrix Inequality for the Inversions of the Restrictions of a Positive Definite Hermitian Matrix

We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.

A NOTE ON COMPARISON THEOREM OF TWO-STAGE ITERATIVE METHOD FOR HERMITIAN POSITIVE DEFINITE LINEAR SYSTEMS

We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.

河套灌区地表水污染溯源分析

针对河套灌区地表水污染来源不清、贡献不明的问题,本文基于9个监测断面月度水质数据,分析灌区地表水环境特征,引用正定矩阵因子分析模型解析污染来源及其空间差异性.结果表明:(1)有机污染指标(COD_(Mn)、COD、BOD_(5))和总氮是河套灌区的主要超标因子,总氮指标低于GB 3838-2002《地表水环境质量标准》Ⅲ类标准的样品数量占比超过67%,整体来说支排干水质差于总排干.其中,七排干入总排干口污染最重,COD_(Mn)与总磷浓度分别超GB 3838-2002Ⅲ类水质标准限值的109%和160%;其次是五排干入总排干口和三排干入总排干口.(2)散养畜禽粪污和秸秆还田以及生产生活污水输入分别是河套灌区地表水有机污染和氮的主要来源,贡献率分别达56.22%和47.82%.(3)河套灌区地表水污染来源具有明显的空间异质性,种植业磷肥输入是影响二排干、三排干和总排干上游水质的主要因素,生产和生活污水输入是影响五排干、七排干和总排干下游水质的主要因素.本文依托监测断面常规监测指标结合数学模型方法,实现了对河套灌区沟渠系统地表水污染的定量溯源,并分析了各监控断面的主要污染风险.

Some Opinions on the Paper “Several Inequalities of Matrix Traces”

In this paper,we find some mistakes in the paper “Several Inequalities of Matrix Traces” which was published in Chinese Quarterly Journal of Mathematics,Vol.10,No.2.

舟山近岸海域沉积物中重金属分布与生态风险评估

为进一步摸清舟山近岸海域表层沉积物重金属的分布特征及污染来源基础情况,2018年8月对舟山近岸海域29个表层沉积物中7种重金属(Cu、Zn、Pb、Cd、Cr、Hg、As)分布特征进行了调查,采用正定因子矩阵(PMF)探究舟山表层沉积物重金属污染来源,并基于潜在生态风险指数法评价了重金属的污染水平及潜在生态风险程度。结果表明:(1)各重金属含量的均值大小排列为Zn>Cr>Cu>Pb>As>Cd>Hg,分布大致由西往东含量逐渐降低,但部分区域存在高值区。各重金属的空间波动程度依次为Pb>Cu>Cd>Cr>Zn>As>Hg。(2) PMF来源解析模型分析出3种主要污染源,分别为交通运输活动、工业生产活动、农业生产活动。(3)各重金属的潜在生态风险系数大小排列为Cd>Hg>Cu>Pb>As>Cr>Zn,其中Cd和Hg为舟山近岸海域的主要潜在生态风险因子。舟山近岸海域沉积物重金属分布受人类活动影响显著,需要开展更全面的风险评价以保护海洋环境和生态健康。

一个线性代数问题的分析巧解

本文的目的是用微积分的方法处理一个判定矩阵正定性的问题.

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.

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.

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.

On Eigenvalues Locations of Symmetric Matrix Families

It is proved that the set of all symmetric real matrices of order n with eigenvalues lying in the interval(α, β), denoted by Sn(α,β), is convex in Rn×n. With this result, some known results on positive(negative) definiteness, and Hurwitz(Shur) stability, as well as the aperiodic property of polytopes of symmetric matrices are generalized, and a series of insightful necessary and sufficient conditions for some general set of symmetric matrices contained in Sn(α,β) are presented,which are directly available for analysis of the positive(negative) definiteness, Hurwitz(Shur) stability and the aperiodic property of a wide class of sets of symmetric matrices.

On the Nonlinear Matrix Equation X＋A~＊f_1（X）A＋B~＊f_2（X）B=Q

In this paper, nonlinear matrix equations of the form X ＋ A＊f1 （X）A ＋ B＊f2 （X）B = Q are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.

Generalizations of a Matrix Inequality

In this paper, some new generalizations of the matrix form of the Brunn-Minkowski inequality are presented.

Analysis of Sparse Quasi-Newton Updates with Positive Definite Matrix Completion

Based on the idea of maximum determinant positive definite matrix completion,Yamashita(Math Prog 115(1):1–30,2008)proposed a new sparse quasi-Newton update,called MCQN,for unconstrained optimization problems with sparse Hessian structures.In exchange of the relaxation of the secant equation,the MCQN update avoids solving difficult subproblems and overcomes the ill-conditioning of approximate Hessian matrices.However,local and superlinear convergence results were only established for the MCQN update with the DFP method.In this paper,we extend the convergence result to the MCQN update with the whole Broyden’s convex family.Numerical results are also reported,which suggest some efficient ways of choosing the parameter in the MCQN update the Broyden’s family.

Hermitian Positive Definite Solutions of the Matrix Equation X ＋ A^＊X^-qA = Q （q≥1）

In this paper,Hermitian positive definite solutions of the nonlinear matrix equation X + A*X-qA = Q(q ≥ 1) are studied.Some new necessary and sufficient conditions for the existence of solutions are obtained.Two iterative methods are presented to compute the smallest and the quasi largest positive definite solutions,and the convergence analysis is also given.The theoretical results are illustrated by numerical examples.

Optimal Bounds for the Largest Eigenvalue of a 3 ×3 Correlation Matrix

A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more stringent than the optimal bounds by Wolkowicz and Styan in specific cases.

Modulus-Based Matrix Splitting Iteration Methods for a Class of Stochastic Linear Complementarity Problem

For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix, respectively. The advantages of the new methods are that they can solve the large scale stochastic linear complementarity problem, and spend less computational time. Numerical results show that the new methods are efficient and suitable for solving the large scale problems.