A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

CONTINUITY OF GENERALIZED INVERSES OF LINEAR OPERATORS IN BANACH SPACES AND ITS APPLICATIONS

The perturbation problem of generalized inverse is studied. And some new stability characteristics of generalized inverses were presented. It was also proved that the stability characteristics of generalized inverses were independent of the choice of the generalized inverse. Based on this result, two sufficient and necessary conditions for the lower semi-continuity of generalized inverses as the set-valued mappings are given.

Unified Approach to the Expression for the Moore-Penrose Inverse of a- xy

Let A be an unital C*-algebra, a, x and y are elements in A. In this paper, we present a method how to calculate the Moore-Penrose inverse of a- xy*and investigate the expression for some new special cases of(a- xy*).

THE METRIC GENERALIZED INVERSE AND ITS SINGLE-VALUE SELECTION IN THE PRICING OF CONTINGENT CLAIMS IN AN INCOMPLETE FINANCIAL MARKET

This article continues to study the research suggestions in depth made by M.Z.Nashed and G.F.Votruba in the journal"Bull.Amer.Math.Soc."in 1974.Concerned with the pricing of non-reachable"contingent claims"in an incomplete financial market,when constructing a specific bounded linear operator A:l_(1)^(n)→l_(2) from a non-reflexive Banach space l_(1)^(n) to a Hilbert space l_(2),the problem of non-reachable"contingent claims"pricing is reduced to researching the(single-valued)selection of the(set-valued)metric generalized inverse A■ of the operator A.In this paper,by using the Banach space structure theory and the generalized inverse method of operators,we obtain a bounded linear single-valued selection A^(σ)=A+of A■.

Position Control of Flexible Joint Carts Using Adaptive Generalized Dynamics Inversion

This paper presents the design and implementation of Adaptive Generalized Dynamic Inversion(AGDI)to track the position of a Linear Flexible Joint Cart(LFJC)system along with vibration suppression of the flexible joint.The proposed AGDI control law will be comprised of two control elements.The baseline(continuous)control law is based on principle of conventional GDI approach and is established by prescribing the constraint dynamics of controlled state variables that reflect the control objectives.The control law is realized by inverting the prescribed dynamics using dynamically scaledMoore-Penrose generalized inversion.To boost the robust attributes against system nonlinearities,parametric uncertainties and external perturbations,a discontinuous control law will be augmented which is based on the concept of sliding mode principle.In discontinuous control law,the sliding mode gain is made adaptive in order to achieve improved tracking performance and chattering reduction.The closed-loop stability of resultant control law is established by introducing a positive define Lyapunov candidate function such that semi-global asymptotic attitude tracking of LFJC system is guaranteed.Rigorous computer simulations followed by experimental investigation will be performed on Quanser’s LFJC system to authenticate the feasibility of proposed control approach for its application to real world problems.

矩阵的Moore-Penrose逆及其应用

由普通的逆矩阵推广到广义逆矩阵,进而研究广义逆矩阵中的Moore-Penrose逆.在矩阵分解的基础上,给出了任意矩阵的Moore-Penrose逆的计算方法,讨论了矩阵的Moore-Penrose逆在求解线性矩阵方程和线性方程组中的应用.

Extremal ranks of the solution to a system of real quaternion matrix equations

In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.

EXTENDED OBLIQUE PROJECTION METHOD FOR GENERALIZED LEAST SQUARES PROBLEM

We extend the oblique projection method given by Y.Saad to solve the generalized least squares problem. The corresponding oblique projection operator is presented and the convergence theorems are proved. Some necessary and sufficient conditions for computing the solution or the minimum N-norm solution of the min || A x- b ||M2 have been proposed as well.

PCR ALGORITHM FOR PARALLEL COMPUTING MINIMUM-NORM LEAST-SQUARES SOLUTION OF INCONSISTENT LINEAR EQUATIONS

This paper presents a new highly parallel algorithm for computing the minimum-norm least-squares solution of inconsistent linear equations Ax = b(A∈Rm×n,b∈R (A)). By this algorithm the solution x = A + b is obtained in T = n(log2m + log2(n - r + 1) + 5) + log2m + 1 steps with P=mn processors when m × 2(n - 1) and with P = 2n(n - 1) processors otherwise.

Common least-squares solution to some matrix equations

Necessary and sufficient conditions are derived for some matrix equations that have a common least-squares solution.A general expression is provided when certain resolvable conditions are satisfied.This research extends existing work in the literature.

SOLUTIONS TO THE SYSTEM OF OPERATOR EQUATIONS AXB = C = BXA

In this paper, we present some necessary and sufficient conditions for the ex- istence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB = C = BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.

ON THE GENERALIZED RESOLVENT OF LINEAR PENCILS IN BANACH SPACES

. Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvent of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil λ→ T - λS are provided and an explicit expression of the generalized resolvent is also given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.

Some Properties of the Optimal Preconditioner and the Generalized Superoptimal Preconditioner

The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively. They have been studied widely since then. Recently, Chen and Jin [6] extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse. In this paper, we further study some useful properties of the optimal and the generalized superoptimal preconditioners. Several existing results are extended and new properties are developed.

EPr Solution to a System of Matrix Equations

A square complex matrix is called if it can be written in the form with being fixed unitary and being arbitrary matrix in . We give necessary and sufficient conditions for the existence of the solution to the system of complex matrix equation and present an expression of the solution to the system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least square solution with least norm to this system mentioned above is considered. The representation of such solution is also derived.

CK引力模型的研究及实证分析--基于Moore-Penrose广义逆测算

文中通过建立地区间CK引力模型,利用最小二乘法确定参数cis和kit。在理论上验证关于未知量cis和kit的线性方程组的解不唯一,并通过Moore-Penrose广义逆求极小范数解,给出求Moore-Penrose广义逆的方法,最后以珠江流域四省市投入产出表为例进行实证分析,结果有效,且计算简便快捷,从而验证该方法切实可行。

The generalized solution of ill-posed boundary problem

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces Wk0,p(Ω, Σ). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.

Perturbations of Moore–Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper, the perturbations of the Moore-Penrose metric generalized inverses of linear operators in Banach spaces are described. The Moore Penrose metric generalized inverse is homo- geneous and nonlinear in general, and the proofs of our results are different from linear generalized inverses. By using the quasi-additivity of Moore-Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition, we show some error estimates of perturbations for the single- valued Moore-Penrose metric generalized inverses of bounded linear operators. Furthermore, by means of the continuity of the metric projection operator and the quasi-additivity of Moore-Penrose metric generalized inverse, an expression for Moore-Penrose metric generalized inverse is given.

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore-Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore-Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore-Penrose metric generalized inverse of bounded linear operators in Banach spaces.

CRITERION OF POSITIVE DEFINITENESS OF MATRICES AND SOLUTION OF INVERSE PROBLEM FOR SYSTEM OF LINEAR EQUATIONS

The class of symmetric definitely positive matrices is extremely important in the matrix theory. At present, positive definiteness of a symmetric matrix can be shown by determining the signs of its all ordered principal minors or the signs

用抛物线Radon变换稀疏解分离和压制多次波

抛物线Radon变换法是目前最常用的、效果较好的一种压制多次波方法。但由于传统的阻尼最小二乘解受分辨率的限制,当动校时差差别较小时,在变换后的τ-q域不能完全分离一次波与多次波,从而不能得到满意的结果。为此,本文提出了一种改进方法,即在最小平方意义下的τ-q域通解中,找出一种沿q值方向具有最大方差模的稀疏解,从而提高了用抛物线Radon变换进行反射波分解的分辨率,更精确、更有效地实现了多次波的分离与压制。与传统的方法相比,在一次波与多次波叠加速度差异较小、多次波能量很强的情况下,本方法表现出较大的优越性。理论数据试算和实际资料处理结果均证实了该方法的有效性。