Research On Core Partial Order And Parallel Sum Of Matrix

Matrix generalized inverse and matrix partial order are the main tools for discussing matrix problems,and they are also one of the important mathematical tools for solving practical problems.They provide abundant resources for matrix theory research.Matrix partial order is a small branch of matrix theory,but It is connected with cybernetics,quantum mechanics,etc.,and promotes the further development of other disciplines,and occupies an important position in the entire matrix theory.Parallel sum is the prelude to the study of the generalized inverse of matrix and the many applications of matrix partial order,and it is the prelude to the matrix partial order theory.Perfection plays an extremely important role.Based on the research theories of various generalized inverses and partial orders of matrices,this article further explores their properties,the inverses and properties of parallel sums,etc.This article is divided into three chapters,the main contents are as follows:The first chapter introduces some definitions and lemmas used in this article.Such as:1-inverse of matrix,index,Moore-Penrose generalized inverse,group inverse,nuclear inverse,parallel sum definition,several related partial ordering Definition,and related lemmas used in the text.The second chapter studies the related properties of the matrix kernel inverse and the equivalent characterization of the kernel partial order.Using matrix decomposition and generalized inverse as tools,the relationship between matrix kernel inverse and group inverse,Moore-Penrose generalized inverse,and the generalized inverse of Moore-Penrose are studied.The nature of the nuclear inverse.On this basis,a matrix partial order kernel partial order based on the nuclear inverse is introduced,and according to some properties of minus partial order,star partial order,and sharp partial order given in the literature,the nuclear partial order is described.The nature of partial order.The related definitions and properties of nuclear inverse and nuclear partial order are explored,which enriches the content of matrix partial order.The third chapter mainly gives a proof of the inverse of a parallel sum.In the literature,some key steps are ignored for its proof.The content of this chapter makes up for the shortcomings of the documentary proof,and explores some properties of parallel Parallel sum is connected with matrix partial order,matrix trace and determinant,etc.,and further improves the matrix partial order theory.