共查询到20条相似文献,搜索用时 31 毫秒
1.
We develop a constructive procedure for generating nonsingular solutions of the matrix equation XA=ATX and establish an interesting relationship between a given solution X of the above equation and the associated matrix polynomial p(A). The latter is then used to develop an algorithm for computing the inertia of a matrix. The algorithm is more efficient than the other common procedures. 相似文献
2.
Based on the elegant properties of the Thompson metric, we prove that the general nonlinear matrix equation Xq-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. 相似文献
3.
In this paper, the nonlinear matrix equation X + A∗XqA = 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. 相似文献
4.
Martine C.B. Reurings 《Linear algebra and its applications》2006,418(1):292-311
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 = In + 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. 相似文献
5.
Suppose that p(X, Y) = A − BX − X(∗)B(∗) − CYC(∗) and q(X, Y) = A − BX + X(∗)B(∗) − CYC(∗) are quaternion matrix expressions, where A is persymmetric or perskew-symmetric. We in this paper derive the minimal rank formula of p(X, Y) with respect to pair of matrices X and Y = Y(∗), and the minimal rank formula of q(X, Y) with respect to pair of matrices X and Y = −Y(∗). As applications, we establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed. 相似文献
6.
7.
Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below:
- (i)
- There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.
8.
Jing Cai 《Applied mathematics and computation》2010,217(1):117-4466
Nonlinear matrix equation Xs + 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. 相似文献
9.
Štefko Miklavi? 《Linear algebra and its applications》2009,430(1):251-636
Let Γ denote a distance-regular graph with diameter D?3. Assume Γ has classical parameters (D,b,α,β) with b<-1. Let X denote the vertex set of Γ and let A∈MatX(C) denote the adjacency matrix of Γ. Fix x∈X and let A∗∈MatX(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of MatX(C) generated by A,A∗. We call T the Terwilliger algebra of Γ with respect to x. We show that up to isomorphism there exist exactly two irreducible T-modules with endpoint 1; their dimensions are D and 2D-2. For these T-modules we display a basis consisting of eigenvectors for A∗, and for each basis we give the action of A. 相似文献
10.
Jian YuDingtao Peng Shuwen Xiang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(17):6326-6332
Let X be a nonempty, convex and compact subset of normed linear space E (respectively, let X be a nonempty, bounded, closed and convex subset of Banach space E and A be a nonempty, convex and compact subset of X) and f:X×X→R be a given function, the uniqueness of equilibrium point for equilibrium problem which is to find x∗∈X (respectively, x∗∈A) such that f(x∗,y)≥0 for all y∈X (respectively, f(x∗,y)≥0 for all y∈A) is studied with varying f (respectively, with both varying f and varying A). The results show that most of equilibrium problems (in the sense of Baire category) have unique equilibrium point. 相似文献
11.
In this paper we study the class of square matrices A such that AA† − A†A is nonsingular, where A† stands for the Moore-Penrose inverse of A. Among several characterizations we prove that for a matrix A of order n, the difference AA† − A†A is nonsingular if and only if R(A)⊕R(A∗)=Cn,1, where R(·) denotes the range space. Also we study matrices A such that R(A)⊥=R(A∗). 相似文献
12.
Consider the nonlinear matrix equation X?=?Q?+?A H (I???X???C) ?? A ( ???=???1 or 0?<?|??|?<?1), where Q is an n×n positive definite matrix, C is an mn ×mn positive semidefinite matrix, I is an m×m identity matrix, and A is an arbitrary mn×n matrix. This equation is connected with a certain interpolation problem when ???=???1. Using the properties of the Kronecker product and the theory for the monotonic operator defined in a normal cone, we prove the existence and uniqueness of the positive definite solution which is contained in the set {X|I???X?>?C} under the condition that I???Q?>?C. The iterative methods to compute the unique solution is proposed. Numerical examples show that the methods are feasible and effective. 相似文献
13.
Antonio S. Granero 《Journal of Mathematical Analysis and Applications》2007,326(2):1383-1393
If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ2-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w∗-compact subset K⊂X∗∗ then and, if K∩C is w∗-dense in K, then . 相似文献
14.
Yongge Tian 《Linear algebra and its applications》2010,433(1):263-296
The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix. 相似文献
15.
The left and right inverse eigenvalue problems of generalized reflexive and anti-reflexive matrices 总被引:1,自引:0,他引:1
Let n×n complex matrices R and S be nontrivial generalized reflection matrices, i.e., R∗=R=R−1≠±In, S∗=S=S−1≠±In. A complex matrix A with order n is said to be a generalized reflexive (or anti-reflexive ) matrix, if RAS=A (or RAS=−A). In this paper, the solvability conditions of the left and right inverse eigenvalue problems for generalized reflexive and anti-reflexive matrices are derived, and the general solutions are also given. In addition, the associated approximation solutions in the solution sets of the above problems are provided. The results in present paper extend some recent conclusions. 相似文献
16.
Xiaoli Zhang 《Journal of Mathematical Analysis and Applications》2008,346(1):251-254
Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. For A,B∈B(H), the Jordan elementary operator UA,B is defined by UA,B(X)=AXB+BXA, ∀X∈B(H). In this short note, we discuss the norm of UA,B. We show that if dimH=2 and ‖UA,B‖=‖A‖‖B‖, then either AB∗ or B∗A is 0. We give some examples of Jordan elementary operators UA,B such that ‖UA,B‖=‖A‖‖B‖ but AB∗≠0 and B∗A≠0, which answer negatively a question posed by M. Boumazgour in [M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386-393]. 相似文献
17.
Yuan Gong Sun 《Journal of Mathematical Analysis and Applications》2003,279(2):651-658
Some new oscillation criteria are established for the matrix linear Hamiltonian system X′=A(t)X+B(t)Y, Y′=C(t)X−A∗(t)Y under the hypothesis: A(t), B(t)=B∗(t)>0, and C(t)=C∗(t) are n×n real continuous matrix functions on the interval [t0,∞), (−∞<t0). These results are sharper than some previous results even for self-adjoint second order matrix differential systems. 相似文献
18.
19.
This paper investigates the matrix equation Am = dl + λJ, where A is a rational circulant. Here d and λ are rational numbers, I is the identity matrix, and J is the matrix with every entry equal to 1. A necessary and sufficient condition is given for the existence of matrices satisfying this equation. Also, it is shown that there is no nontrivial solution if entries of A are restricted to take only values 0 and 1. 相似文献
20.
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. 相似文献