Core-EP Inverses And Weak Group Inverses

Generalized inverse theory plays an important role in many fields,such as,dif-ferential equations,numerical analysis,electrical network analysis,optimization,Markov chains,and system theory.Moore-Penrose inverses and Drazin inverses are two types of classical gen-eralized inverses.The development of generalized inverses tends to be diversified,resulting in many new generalized inverses.For example,the core inverse,the core-EP inverse,the weak group inverse.Based on high speed computing capability of neural networks,many papers have provided different types of recurrent neural networks to calculate generalized inverses of high-order matrices.This paper is devoted to the research of core-EP inverses,weak group inverses and calcu-lation of core-EP inverses of time-varying complex matrices based on recurrent neural network.Firstly,Chapter 2 studies the core invertibility of the linear combination of two core in-vertible complex matrices P,Q under conditions P2P(?)Q(?)Q=QPP(?)and QP(?)P=P,respectively.And gives the corresponding expression of the core inverse.It generalize relevant results of Xiaoji Liu et al.which relate to the group invertibility of the linear combination of two group invertible complex matrices.Finally,the core invertibility of the sum and difference of two core invertible elements in rings with involution is investigated,and the corresponding expression is given.Chapter 3 investigates two types of integral representations of the core-EP inverse firstly.The first type is based on the full-rank decomposition of a given matrix.The second type is based on the expression of the core-EP inverse.These two integral representations do not have any restrictions on the spectrum of matrices.Secondly,limit representations of three types of core-EP inverses are given,which are based on the full-rank decomposition of a given matrix and the expression of the core-EP inverse.In particular,integral and limit representations of the core inverse are given.Chapter 4 first describes the pseudo-core inverse by projection,and gives a corresponding pseudo core inverse expression.Then,it is proved that the pseudo core inverse of an element is an EP element,and a new expression of the pseudo core inverse is given.The relationship between EP elements and pseudo-core inverses is studied.It generalizes the relevant result of Sanzhang Xu et al.which studies core inverses.Finally,expressions of pseudo core inverses are obtained by Pierce decomposition and inner inverses,it extends corresponding results of Ferreyra et al.which characterize representations of the core-EP inverse by using inner inverses.Chapter 5 is mainly based on the recurrent neural network to compute the core inverse and the core-EP inverse of a time-varying complex matrix.Firstly,an improved complex varying-parameter Zhang neural network model(CVPZNN)is established to compute the outer inverse.Two kinds of Zhang neural network models to calculate the the core-EP inverse of a time-varying complex matrix are given.The convergence rate of Zhang neural network model is accelerated.The super-exponential performance of CVPZNN with a linear activation function is proved.Then the upper bound of the finite time convergence of CVPZNN with Li activation function and CVPZNN with tunable activation function are estimated,respectively.Finally,simulation results of CVPZNNs with different activation functions are given.Chapter 6 mainly studies the weak group inverse of an element in a proper*-ring.Firstly,the concept of the weak group inverse of the complex matrix proposed by Hongxing Wang and Jianlong Chen is extended to the proper*-ring by three equations.The necessary and sufficient conditions for the existence of the weak group inverse are given.The weak group inverse is characterized by an idempotent element,and the corresponding expression of the weak group inverse is given.The reverse order law and addition property of the weak group inverse are proved under certain conditions.Secondly,the group-EP decomposition are defined.By group-EP decomposition,some new properties of the weak group inverse are obtained.Equivalent characterizations of a(?)a*=a*a(?)are given by using the normality of the group invertible part of an element in its group-EP decomposition.Finally,the weak group element is defined in a proper*-ring and some equivalent characterizations of weak group elements are given by using the Drazin inverse and {1,3}-inverses.