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.     

GLOBAL EXISTENCE AND EXPONENTIAL STABILITY OF SOLUTIONS TO THE QUASILINEAR THERMO-DIFFUSION EQUATIONS WITH SECOND SOUND

This article is devoted to the study of global existence and exponential stability of solutions to an initial-boundary value problem of the quasilinear thermo-diffusion equations with second sound by means of multiplicative techniques and energy method provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.     

正定性与二次函数的极值

In this paper we give out a sufficient and solution of quadratic function's maximum with theory of quadratic form and give out the definition of the positive definite property of the following homogeneous polynomials of degree 2n two variables function, based on the definite of local maximum of two variables function.     

ON SOLUTIONS OF MATRIX EQUATION AXAT ＋ BYBT=C

By making use of the quotient singular value decomposition (QSVD) of a matrix pair,this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the general solutions of the linear matrix equation AXA^T BYB^T=C with the unknown X and Y, which may be both symmetric, skew-symmetric, nonnegativede finite, positive definite or some cross combinations respectively. Also, the solutions of some optimal problems are derived.     

ON LIMIT CYCLES FOR A CLASS OF DIFFERENTIAL SYSTEMS WITH POSITIVE DEFINITE POLYNOMIAL

In this paper, the nonexistence, existence and the number of limit cycles for a class of differential systems with positive definite polynomial are considered, and the results obtained generalize and supplement those of [1].     

ON NEWTON-HSS METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS WITH POSITIVE-DEFINITE JACOBIAN MATRICES

The Hermitian and skew-Hermitian splitting （HSS） method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implemen- tations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step.     

Complementarity properties of the Lyapunov transformation over symmetric cones

The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX +XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V . Given any element a ∈ V , we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0 -property for La and show that La has the R0 -property if and only if a is invertible. Finally, we provide La with some characterizations of the E0 -property and the nondegeneracy property.     

On Development of a Ky Fan Inequality of the Complementary A-G Type

Two new proofs of a discrete Ky Fan inequality of the complementary A-G type are given. Its continuous version and determinantal analogueon a set of pairwise commutative positive definite matrices are establishbed. Afurther extension concerning general positive definite matrices of the inequalityis also suggested.     

Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.     

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.     

矩阵方程X+A^{*}X^{-q}A=Q(q\geq 1)的Hermitian正定解

本文研究矩阵方程X A~*X~(-q)A=Q(q≥1)的Hermitian正定解,给出了存在正定解的充分条件和必要条件,构造了求解的迭代方法.最后还用数值例子验证了迭代方法的可行性和有效性.     

Some iterative algorithms for positive definite solution to nonlinear matrix equations

This paper is concerned with the unique positive definite solution to a system of nonlinear matrix equations $X-A^*\bar{Y}^{-1}A=I_n$ and $Y-B^*\bar{X}^{-1}B=I_n$, where $A,B\in\mathbb{C}^{n\times n}$ are given matrices. Based on the special structure of the system of nonlinear matrix equations, the system can be equivalently reformulated as $V-C^*\bar{V}^{-1}C=I_{2n}$. Moreover, by means of Sherman-Moorison-Woodbury formula, we derive the relationship between the solutions of $V-C^*\bar{V}^{-1}C =I_{2n}$ and the well studied standard nonlinear matrix equation $Z+D^*Z^{-1}D=Q$, where $D$, $Q$ are uniquely determined by $C$. Then, we present a structure-preserving doubling algorithm and two modified structure-preserving doubling algorithms to compute the positive definite solution of the system. Furthermore, cyclic reduction algorithm and two modified cyclic reduction algorithms for the positive definite solution of the system are proposed. Finally, some numerical examples are presented to illustrate the efficiency of the theoretical results and the behavior of the considered algorithms.     

Norm estimations for perturbations of the weighted Moore-Penrose inverse

For a complex matrix $A\in \mathbb{C}^{m\times n}$, the relationship between the weighted Moore-Penrose inverse $A^\dag_{M_1N_1}$ and $A^\dag_{M_2N_2}$ is studied, and an important formula is derived,where $M_1\in \mathbb{C}^{m\times m}, N_1\in\mathbb{C}^{n\times n}$ and $M_2\in \mathbb{C}^{m\times m}, N_2\in\mathbb{C}^{n\times n}$ are different pair of positive definite hermitian matrices. Based on this formula, this paper initiates the study of the perturbation estimations for $A^\dag_{MN}$ in the case that $A$ is fixed, whereas both $M$ and $N$ are variable. The obtained norm upper bounds are then applied to the perturbation estimations for the solutions to the weighted linear least squares problems.     

On extensions of positive definite integral Fredholm operators

The main result of this paper states that a positive definite Fredholm integral operator acting on L²([0,1]) can be modified on a Lebesque measurable set D\mit\Delta in [0,1]² such that the resulting operator is positive definite and its resolvent kernel is zero on D\mit\Delta . This answers a question raised in [3]. The proof is based on extension results for positive definite operator matrices and their connection to generalized determinants.     

非自治系统渐近稳定的充分条件

本文对 Liapunov关于渐近稳定性的基本定理进行改进 ,在附加辅助函数前提下 ,同时放宽了 V正定 ,有无穷小上界 .V负定的条件 .改进了文 [1 ]～ [6]相应结果     

On the number of spanning trees in graphs with multiple edges

In this paper, we consider a class of nonlinear matrix equation of the type \(X+\sum _{i=1}^mA_i^{*}X^{-q}A_i-\sum _{j=1}^nB_{j}^{*}X^{-r}B_j=Q\), where \(0<q,\,r\le 1\) and Q is positive definite. Based on the Schauder fixed point theorem and Bhaskar–Lakshmikantham coupled fixed point theorem, we derive some sufficient conditions for the existence and uniqueness of the positive definite solution to such equations. An iterative method is provided to compute the unique positive definite solution. A perturbation estimation and the explicit expression of Rice condition number of the unique positive definite solution are also established. The theoretical results are illustrated by numerical examples.     

ON THE CONSTRUCTION OF INDECOMPOSABLE POSITIVE DEFINITE QUADRATIC FORMS OVER Z

The aim of this paper is to construct indecomposable positive definite Z-lattices withgiven rank n(2≤n≤9) and discriminant a,a being composite and square-free,but for afinite number of exceptions.And all exceptional cases for 3≤n≤9 are determined.Theseare unsettled cases of a paper of O’Meara in 1975.     

EXISTENCE CONDITIONS OF A SPECIAL TYPE OF LIAPUNOV FUNCTIONAL FOR TWO-DIMENSIONAL DELAY SYSTEM

In this paper, the author considers the two-dimensional delay systems $$\[\mathop x\limits^ \cdot (t) = Ax(t) + Bx(t - r),A,B \in {R^{2 \times 2}},x \in {R^2},r = const \ge 0\]$$ and gives the necessary and sifficient conditions under which where exists a simple type of positive definite Liapunov functional $$\[V(\varphi ) \buildrel \Delta \over = {\varphi ^''}(0){T_\varphi }(0) + \int_{ - \tau }^0 {{\varphi ^''}(\theta )E\varphi (\theta )d\theta } \]$$ and $\[\alpha (s)\]$(where T , E are positive definite 2x2 matrices, $\[\varphi \in C([ - \tau ,0],{R^n})\]$, "." stands for transpose, $\[\alpha (s)\]$ is continuous and $\[\alpha (0) = 0,\alpha (s) > 0,s > 0\]$. such that $\[{V_{(*)}}(\varphi ) \le - \alpha (\left| {\varphi (0)} \right|).\]$.     

矩阵方程aX2+bX+cE=O的正定解和实对称解

给出了矩阵方程aX2+bX+cE=O,a,b,c∈R,a≠0有正定解,实对称解的充分必要条件及解的一般形式.