环上矩阵的加权广义逆

该文从另一方面给出了Hartwig R E提出的公开问题的解答.设R是带有对合*的任意环,作者定义了环R上的一种新的加权广义逆,记为A_(P,Q)~+,并给出了A_(P,Q)~+存在的充要条件.同时,得到了一些重要的性质.     

The error bound of the perturbation of the Drazin inverse

Let A and E be n×n matrices and B = A + E. Denote the Drazin inverse of A by A^D. In this paper we give an upper bound for the relative error ∥B^D ? A^D∥/∥A^D∥₂ and a lower bound for ∥B^D∥₂ under certain circumstances. The continuity properties and the derivative of the Drazin inverse are also considered.     

On the properties of generalized frames

In this paper, we introduce the notion of generalized frames and study their properties. Discrete and integral frames are special cases of generalized frames. We give criteria for generalized frames to be integral (discrete). We prove that any bounded operator A with a bounded inverse acting from a separable space H to L ₂(Ω) (where Ω is a space with countably additive measure) can be regarded as an operator assigning to each element x ∈ H its coefficients in some generalized frame.     

The computation and perturbation analysis for weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29(1):39–48, 2007) has defined a weighted group inverse of rectangular matrices. Let A∈C ^m×n ,W∈C ^n×m , if X∈C ^m×n satisfies the system of matrix equations $$(W_{1})\quad AWXWA=A,\quad\quad (W_{2})\quad XWAWX=X,\quad\quad (W_{3})\quad AWX=XWA$$ X is called the weighted group inverse of A with W, and denoted by A _W ^# . In this paper, we will study the algebra perturbation and analytical perturbation of this kind weighted group inverse A _W ^# . Under some conditions, we give a decomposition of B _W ^# ?A _W ^# . As a results, norm estimate of ‖B _W ^# ?A _W ^# ‖ is presented (where B=A+E). An expression of algebra of perturbation is also obtained. In order to compute this weighted group inverse with ease, we give a new representation of this inverse base on Gauss-elimination, then we can calculate this weighted inverse by Gauss-elimination. In the end, we use a numerical example to show our results.     

Some refined bounds for the perturbation of the orthogonal projection and the generalized inverse

In this paper, we consider the perturbation of the orthogonal projection and the generalized inverse for an n × n matrix A and present some perturbation bounds for the orthogonal projections on the rang spaces of A and A^?, respectively. A combined bound for the orthogonal projection on the rang spaces of A and A^? is also given. The proposed bounds are sharper than the existing ones. From the combined bounds of the orthogonal projection on the rang spaces of A and A^?, we derived new perturbation bounds for the generalized inverse, which always improve the existing ones. The combined perturbation bound for the orthogonal projection and the generalized inverse is also given. Some numerical examples are given to show the advantage of the new bounds.     

Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities

In 1956, R. Penrose studied best-approximate solutions of the matrix equation AX = B. He proved that A⁺B (where A⁺ is the Moore-Penrose inverse) is the unique matrix of minimal Frobenius norm among all matrices which minimize the Frobenius norm of AX ? B. In particular, A⁺ is the unique best-approximate solution of AX = I. The vector version of Penrose's result (that is, the fact that the vector A⁺b is the best-approximate solution in the Euclidean norm of the vector equation Ax = b) has long been generalized to infinite dimensional Hilbert spaces.In this paper, an infinite dimensional version of Penrose's full result is given. We show that a straightforward generalization is not possible and provide new extremal characterizations (in terms of the Hermitian order) of A⁺ and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least-squares solutions. For a certain class of operators, we can phrase our characterizations in terms of a whole class of norms (including the Hilbert-Schmidt and the trace norms), thus providing new extremal characterizations even in the matrix case. We treat both operators with closed range and with not necessarily closed range. Finally, we characterize A⁺ as the unique inner inverse of minimal Hilbert-Schmidt norm if ∥A⁺∥₂ < ∞. We give an application of the new extremal characterization to the compensation problem in systems analysis in infinite-dimensional Hilbert spaces.     

Drazin inverse of partitioned matrices in terms of Banachiewicz-Schur forms

Let be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A^D. The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S=D-CA^DB, group invertible, can be expressed in terms of a matrix in the Banachiewicz-Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rank(M)=rank(A^D)+rank(S^D), and give conditions under which the group inverse of M exists and a formula for its computation.     

On the characterization of generalized inverses by bordered matrices

A method to characterize the class of all generalized inverses of any given matrix A is considered. Given a matrix A and a nonsingular bordered matrix T of A,

$\text{T=}\text{A}\text{P}\text{Q}\text{R}$

the submatrix, corresponding to A, of T^-1 is a generalized inverse of A, and conversely, any generalized inverse of A is obtainable by this method. There are different definitions of a generalized inverse, and the arguments are developed with the least restrictive definition. The characterization of the Moore-Penrose inverse, the most restrictive definition, is also considered.     

Characterizations of EP matrices and weighted-EP matrices

A square complex matrix A is said to be EP if A and its conjugate transpose A^∗ have the same range. In this paper, we first collect a group of known characterizations of EP matrix, and give some new characterizations of EP matrices. Then, we define weighted-EP matrix, and present a wealth of characterizations for weighted-EP matrix through various rank formulas for matrices and their generalized inverses.     

Operator and matrix representation for the generalized inverse AT,S(2)A_{T,S}^{(2)}

This paper presents the matrix representation for extension of inverse of restriction of a linear operator to a subspace, on the basis of which we establish useful representations in operator and matrix form for the generalized inverse A_T,S⁽²⁾A_{T,S}^{(2)} and give some of their applications.     

An oblique projection iterative method to compute Drazin inverse and group inverse

In this paper, an oblique projection iterative method is presented to compute matrix equation AXA=A of a square matrix A with ind(A)=1. By this iterative method, when taken the initial matrix X₀=A, the group inverse A_g can be obtained in absence of the roundoff errors. If we use this iterative method to the matrix equation A^kXA^k=A^k, a group inverse (A^k)_g of matrix A^k is got, then we use the formulae A_d=A^k-1(A^k)_g, the Drazin inverse A_d can be obtained.     

On the Drazin inverse of block matrices and generalized Schur complement

In this paper, we give an additive result for the Drazin inverse with its applications, we obtain representations for the Drazin inverse of a 2 × 2 complex block matrix having generalized Schur complement S=D-CA^DB equal to zero or nonsingular. Several situations are analyzed and recent results are generalized [R.E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (3) (2006) 757-771].     

The left and right inverse eigenvalue problems of generalized reflexive and anti-reflexive matrices

Let n×n complex matrices R and S be nontrivial generalized reflection matrices, i.e., R^∗=R=R⁻¹≠±I_n, S^∗=S=S⁻¹≠±I_n. 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.     

On the relation of closed forms and Trotter's product formula

The aim of this paper is to give a characterization in Hilbert spaces of the generators of C₀-semigroups associated with closed, sectorial forms in terms of the convergence of a generalized Trotter's product formula. In the course of the proof of the main result we also present a similarity result which can be of independent interest: for any unbounded generator A of a C₀-semigroup e^tA it is possible to introduce an equivalent scalar product on the space, such that e^tA becomes non-quasi-contractive with respect to the new scalar product.     

Generalized inverses on normed vector spaces

We extend the concepts, introduced by C.R. Rao for Euclidean norms, of minimum g-inverses and least square g-inverses, using arbitrary norms. We give a characterization of such generalized inverses and an application to the case in which the norm is l_∞. As a result of this application we obtain that when $\text{A?}\text{C}{}^{\mathrm{(n+1)\times n}}$ has rank n, there exists a generalized inverse of A, which serves the same purpose as the Moore-Penrose inverse, when the norm is l_∞.     

Approximations in the l∞ norm and the generalized inverse

We introduce the concept of a strict l_∞-metric projector, based in the definition of strict approximation, to prove that for each matrix A of order m×n with coefficients in the field R of real numbers there exists a set of operators G: R^m → Rⁿ homogeneous and continuous, but not necessarily linear (strict generalized inverse) such that AGA = A and 6AGy?y6 is minimized for all y, when the norm is the l_∞ norm. We investigate the properties of these operators and prove that there are two distinguished operators A^-1_{∞, β} and A^-1_∞ which are extensions of the generalized inverse introduced by Newman and Odell in the case of a strictly convex norm.     

A note on computational formulas for the Drazin inverse of certain block matrices

In this paper, we give a computational formula for the Drazin inverse of the sum P+Q, then applying it we give some computational formulas for the Drazin inverse of block matrix (A and D are square) with generalized Schur complement S=D?CA ^D B is nonsingular under some conditions. These results extend the results about the Drazin inverse of M given by R. Hartwig, X. Li and Y.?Wei (SIAM J. Matrix Anal. Appl. 27:757?C771, 2006) and by C. Deng (J. Math. Anal. Appl. 368:1?C8, 2010).     

The representation and approximations of outer generalized inverses

We present a unified representation theorem for the class of all outer generalized inverses of a bounded linear operator. Using this representation we develop a few specific expressions and computational procedures for the set of outer generalized inverses. The obtained result is a generalization of the well-known representation theorem of the Moore--Penrose inverse as well as a generalization of the well-known results for the Drazin inverse and the generalized inverse A_T,S ⁽²⁾. Also, as corollaries we get corresponding results for reflexive generalized inverses.     

Generalizations of the reverse order law with related results

This paper presents conditions which are necessary and sufficient for (AB>)⁺ = B⁺A^ω for all normalized generalized inverses A^ω of the complex matrix A. Corresponding conditions are stated which are equivalent to the situation where (AB)⁺ = B_ωA⁺ is satisfied by each weak generalized inverse B_ω of B. The results are applied to theorems by Baskett and Katz and by Schwerdtfeger.     

The Schur complement and the inverse M-matrix problem

It is well known that if M is a nonnegative nonsingular inverse M-matrix and if A is a nonsingular block in the upper left hand corner, then the Schur complement of A in M, (M ? A), is an inverse M-matrix. The converse of this is generally false. In this paper we give added restrictions on M or A to insure the converse, and give some necessary and sufficient conditions for HMH^?1, where H = I⊕(?I), to be an M-matrix.