矩阵方程X-A*X-2A=Q的正定解及其扰动分析

研究非线性矩阵方程X-A*X-2A=Q(Q>0)的Hermite正定解及其扰动问题.给出了该方程存在唯-Hermite正定解的充分条件及解的迭代计算公式.在此条件下,给出了该唯一解的扰动界及正定解条件数的一种表达式,并用数值例子对所得结果进行了说明.     

Perturbation estimates for the nonlinear matrix equation X−A * X q A=Q (0<q<1)

The nonlinear matrix equation X?A ^* X ^q A=Q with 0<q<1 is investigated. Two perturbation estimates for the unique positive definite solution of the equation are derived. The theoretical results are illustrated by numerical examples.     

Positive definite solutions of the nonlinear matrix equation X + AXA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A^∗X^qA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.     

Perturbation analysis of the matrix equation

Consider the nonlinear matrix equation X-A^*X^-pA=Q with 0<p1. This paper shows that there exists a unique positive definite solution to the equation. A perturbation bound and the backward error of an approximate solution to this solution is evaluated. We also obtain explicit expressions of the condition number for the unique positive definite solution. The theoretical results are illustrated by numerical examples.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

Perturbation bounds for the Cholesky andQR factorizations

LetA, A+E be Hermitian positive definite matrices. Suppose thatA=LL ^H andA+E=(L+G)(L+G)^H are the Cholesky factorizations ofA andA+E, respectively. In this paper lower bounds and upper bounds on |G|/|L| in terms of |E|/|A| are given. Moreover, perturbation bounds are given for the QR factorization of a complexm ×n matrixA of rankn.This research was supported by the National Science Foundation of China and the Department of Mathematics of Linköping University in Sweden.     

Thompson metric method for solving a class of nonlinear matrix equation

Based on the elegant properties of the Thompson metric, we prove that the general nonlinear matrix equation X^q-A^∗F(X)A=Q(q>1) always has a unique positive definite solution. An iterative method is proposed to compute the unique positive definite solution. We show that the iterative method is more effective as q increases. A perturbation bound for the unique positive definite solution is derived in the end.     

On two perturbation estimates of the extreme solutions to the equations X ± A*XA = Q

Two perturbation estimates for maximal positive definite solutions of equations X + A*X⁻¹A = Q and X − A*X⁻¹A = Q are considered. These estimates are proved in [Hasanov et al., Improved perturbation Estimates for the Matrix Equations X ± A*X⁻¹A = Q, Linear Algebra Appl. 379 (2004) 113-135]. We derive new perturbation estimates under weaker restrictions on coefficient matrices of the equations. The theoretical results are illustrated by numerical examples.     

On Additive Decomposition of the Hermitian Solution of the Matrix Equation AXA* = B

The decomposition of a Hermitian solution of the linear matrix equation AXA* = B into the sum of Hermitian solutions of other two linear matrix equations A₁X₁A^*₁ = B₁{A_{1}X_{1}A^{*}_{1} = B_{1}} and A₂X₂A^*₂ = B₂{A_{2}X_{2}A^*_{2} = B_{2}} are approached. As applications, the additive decomposition of Hermitian generalized inverse C ⁻ = A ⁻ + B ⁻ for three Hermitian matrices A, B and C is also considered.     

On the Hermitian positive defnite solution of the nonlinear matrix equation X + A*X −1 A + B*X −1 B = I

In this paper, we study the matrix equation X + A*X ⁻¹ A + B*X ⁻¹ B = I, where A, B are square matrices, and obtain some conditions for the existence of the positive definite solution of this equation. Two iterative algorithms to find the positive definite solution are given. Some numerical results are reported to illustrate the effectiveness of the algorithms. This research supported by the National Natural Science Foundation of China 10571047 and Doctorate Foundation of the Ministry of Education of China 20060532014.     

On positive definite solution of a nonlinear matrix equation

In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A^*X^?αA = Q with α ∈ (0, ∞) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Linear preservers and quantum information science

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn?×?mn Hermitian matrices such that φ(A???B) and A???B have the same spectrum for any m?×?m Hermitian A and n?×?n Hermitian B. Such a map has the form A???B???U(?₁(A)????₂(B))U* for mn?×?mn Hermitian matrices in tensor form A???B, where U is a unitary matrix, and for j?∈?{1,?2}, ? _j is the identity map?X???X or the transposition map?X???X ^t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A???B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.     

Multivariate normal distributions,Fisher information and matrix inequalities

Using appropriately parameterized families of multivariate normal distributions and basic properties of the Fisher information matrix for normal random vectors, we provide statistical proofs of the monotonicity of the matrix function A ^-1 in the class of positive definite Hermitian matrices. Similarly, we prove that A ¹¹ &lt; A ^-111, where A ₁₁ is the principal submatrix of A and A ¹¹ is the corresponding submatrix of A ^-1. These results in turn lead to statistical proofs that the the matrix function A ^-1 is convex in the class of positive definite Hermitian matrices and that A ² is convex in the class of all Hermitian matrices. (These results are based on the Loewner ordering of Hermitian matrices, under which A &lt; B if A - B is non-negative definite.) The proofs demonstrate that the Fisher information matrix, a fundamental concept of statistics, deserves attention from a purely mathematical point of view.     

On the nonlinear matrix equation

Based on fixed point theorems for monotone and mixed monotone operators in a normal cone, we prove that the nonlinear matrix equation always has a unique positive definite solution. A conjecture which is proposed in [X.G. Liu, H. Gao, On the positive definite solutions of the matrix equation X^s±A^TX^-tA=I_n, Linear Algebra Appl. 368 (2003) 83–97] is solved. Multi-step stationary iterative method is proposed to compute the unique positive definite solution. Numerical examples show that this iterative method is feasible and effective.     

A new inversion-free method for a rational matrix equation

Motivated by the classical Newton-Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation X+A^∗X^-1A=I, where I is the identity matrix and A is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix AA^∗. We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.     

On a class of matrices which arise in the numerical solution of Euler equations

Summary We study block matricesA=[A_ij], where every blockA _ij ^k,k is Hermitian andA _ii is positive definite. We call such a matrix a generalized H-matrix if its block comparison matrix is a generalized M-matrix. These matrices arise in the numerical solution of Euler equations in fluid flow computations and in the study of invariant tori of dynamical systems. We discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized H-matrix.Research supported by the Graduiertenkolleg mathematik der Universität Bielefeld     

Perturbation bounds for theLDL H andLU decompositions

LetA andA+A be Hermitian positive definite matrices. Suppose thatA=LDL ^H and (A+A)=(L+L)(D+D)(L+L)^H are theLDL ^H decompositons ofA andA+A, respectively. In this paper upper bounds on |D| _F and |L| _F are presented. Moreover, perturbation bounds are given for theLU decomposition of a complexn ×n matrix.     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

On principal submatrices

It is shown that if A[ω] is a principal submatrix of the positive definite Hermitian matrix A, then A ^?1[ω] ?(A[ω])^?1is a positive semidefinite hermitian matrix. This fact is used to give a brief proof of a result of Saburou Saitoh concerning Hadamard products.