Least Squares Properties of Generalized Inverses

The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linear matrix equations (i.e.both least squares and the minimal norm) is considered.Also,characterizations of least squares solution and solution of minimum norm are given.Basic properties of the Drazin-inverse solution and the outer-inverse so-lution are present.Motivated by recent research,important least square prop-erties of composite outer inverses are collected.     

Least Squares Properties of Generalized Inverses

The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linear matrix equations (i.e.both least squares and the minimal norm) is considered.Also,characterizations of least squares solution and solution of minimum norm are given.Basic properties of the Drazin-inverse solution and the outer-inverse so-lution are present.Motivated by recent research,important least square prop-erties of composite outer inverses are collected.     

集合H上矩阵A的左(右)逆、伪左(右)逆

以集合 S与空集Φ的交、并运算为背景 ,定义集合 H ={ 0 ,1 }中的加法与乘法运算 0 ,并考虑 H上一个 s×n级矩阵的左逆矩阵、右逆矩阵以及伪左逆矩阵、伪右逆矩阵的定义 ,并且证明了矩阵 A有左、右逆矩阵 ,A有伪左、右逆矩阵的充分必要条件 .     

Core–EP inverse

In this work, we introduce a notion of ‘core–EP inverse’ for a square matrix which is not essentially of index one. This extends the notion of ‘core inverse’, which was initially defined for the matrices of index one. The properties of matrices having ‘core–EP inverse’ and ‘core–EP generalized inverse’ are studied, and obtained a formula to compute the core–EP generalized inverse from a particular linear combination of minors of given matrix.     

The Pseudo Drazin inverses in Banach Algebras

Let (A) be a complex Banach algebra and J be the Jacobson radical of(A).(1) We firstly show that a is generalized Drazin invertible in (A) if and only if a+J is generalized Drazin invertible in (A)/J.Then we prove that a is pseudo Drazin invertible in (A) if and only if a + J is Drazin invertible in (A)/J.As its application,the pseudo Drazin invertibility of elements in a Banach algebra is explored.(2) The pseudo Drazin order is introduced in (A).We give the necessary and sufficient conditions under which elements in (A) have pseudo Drazin order,then we prove that the pseudo Drazin order is a pre-order.     

Representations and properties of the W-Weighted Drazin inverse

Several new representations of the W-weighted Drazin inverse are introduced. These representations are expressed in terms of various matrix powers as well as in terms of matrix products involving the Moore–Penrose inverse and the usual matrix inverse. Also, the properties of various generalized inverses which arise from derived representations are investigated. The computational complexity and efficiency of the proposed representations are considered. Representations are tested and compared among themselves in a substantial number of randomly generated test examples.     

The group inverse of a product

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.     

Approximation Theorems of Moore-Penrose Inverse by Outer Inverses

1 Introduction and preliminaries Let X and Y be two Hilbert spaces and T a bounded linear operator from X into Y . We use D(T ), N(T ) and R(T ), respectively, to denote the domain, null space and range of T . Recall that a linear operator T # : Y → X is…     

Pseudo core inverses in rings with involution

Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in R. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary ?-ring case by Raki?, Din?i? and Djordjevi? in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index in R. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of ?-ring. Some equivalent characterizations for elements in R to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.     

关于广义逆矩阵AT，S^（1,2）的刻画推广

本文通过一类秩等方程给出了AT,S^(1,2)、AT,S^(2)的一种刻画及一类秩等方程有解的充分必要条件，推广了文献[1]、[4]的结论，并改进了[4]关于矩阵A的Drazin逆Af的一类刻画的证明。     

Banach空间中线性算子的齐性广义逆

本文首先在Banach空间内引进拟线性投影算子的概念,由此给出Banach空 间内线性算子的齐性广义逆的统一定义。齐性广义逆包含线性广义逆、单值度量广义 逆.本文证得齐性广义逆存在的充分必要条件.     

Some results on the Drazin inverse of anti-triangular matrices

Abstract

The representations for the Drazin inverse of anti-triangular matrices are obtained under some conditions. Applying these representations, we give a necessary condition for a class of block matrices to have signed Drazin inverse.     

具有广义分解的态射的广义逆

本文给出了预加范畴中态射的广义分解的概念,并研究了具有广义分解的态射的｛１,ｉ｝－逆,Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ逆存在的条件及其表达式,得到了态射的群逆及Ｄｒａｚｉｎ逆存在的充要条件,推广了具有泛分解的态射的广义逆的相应结果．     

Γ-inverses of Bounded Linear Operators

Let B(H) be the algebra of all the bounded linear operators on a Hilbert space H.For A,P and Q in B(H),if there exists an operator X∈ B(H) such thatAP X QA=A,X QAP X=X,(QAP X)*=QAP X and(X QAP)*=X QAP,then X is said to be the Γ-inverse of A associated with P and Q,and denoted by AP,Q+.In this note,we present some necessary and su?cient conditions for which A+P,Qexists,and give an explicit representation of AP,Q+(if AP,Q+exists).     

Weighted binary relations involving core-EP inverse

We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations. This relation becomes a pre-order. Then, one-sided pre-orders associated to the weighted core-EP inverse are given from two perspectives. Finally, we make a comparison for these two sets of one-sided weighted pre-orders.     

On E-solid Semigroups with Inverse Transversals

We characterize E-solid semigroups with inverse transversals and some special such semigroups in terms of congruences. Some properties of inverse congruences over completely simple semigroups are obtained.1991 MR Subject Classification: 20M10This work was supported by the Natural Science Foundation of Guangdong, China.     

The representations of the Drazin inverse of differences of two matrices

In this paper, we give explicit representations of (P ± Q)^D, (P ± PQ)^D and (PQ)^# of two matrices P and Q, as a function of P, Q, P^D and Q^D, under the conditions P³Q = QP and Q³P = PQ.     

加法范畴中态射的Drazin逆

本文研究了加法范畴上态射的Drazin逆。首先给出了态射和φ η与态射φ有Drazin逆的一个关系，得到了φ η的Drazin逆的一个公式，其次证明了态射φ有Drazin逆当且仅当φ^k有群逆（k为某一正整数）。最后还证明了：如果2为可逆态射，则具有Drazin逆的态射一定为两个可逆态射之和。