| The partial order and orthogonality of matrices are widely used,which involves information,communication,statistics,engineering and many other fields.In recent years,they have been hot topics in the field of matrix research.In 2010,Baksalary and Trenkler proposed the concepts of core inverse and core partial order.In 2021,Ferreyra and Malik proposed the concepts of core orthogonality and strong core orthogonality in group invertible matrices,and pointed out the equivalence relationship between core orthogonality and core partial order.Some results of core partial order and core orthogonality in the set of complex matrices are researched in this paper using the core-EP decomposition.The content of this article is divided into five chapters,and the specific content is as follows:In chapter 1,the research background and development status of the generalized inverses,partial orders and orthogonality of matrices at home and abroad are introduced.In chapter 2,Some relevant symbols,lemmas and other preparatory knowledge are given.In chapter 3,the equivalent characterization of the core partial order is mainly discussed.Firstly,by the core-EP decomposition,the equivalent decomposition forms of the core partial order and the minus partial order are given respectively in the set of group invertible matrices.The form of core inverse about special block matrix is analyzed.Furthermore,the necessary and sufficient condition for the core partial order is derived,which is related to the minus partial order and {1,3}-inverse.This shows that the rank-subtractivity and core-subtractivity properties can result the core partial order,which answers the open question raised in reference[14].Subsequently,some new conclusions of the core partial order are founded,which is about core inverse and group inverse.In chapter 4,the core orthogonality and strong core orthogonality of matrices are studied.Firstly,the concepts of unilateral core orthogonality are defined.Then,the equivalence between core orthogonality and strong core orthogonality is pointed out under the EP matrix.Next,some conclusions about core orthogonality and strong core orthogonality proposed by Ferreyra and Malik are corrected.Furthermore,the equivalence between the strong core orthogonality and rank-additivity,core-additivity is proved,which answers the open question raised in [13].In addition,using the matrix equations,Hermite matrices,other equivalent conditions are given.In Chapter 5,a summary of the current research work has been made,and relevant issues worth discussing in the future have also been raised. |
