Research On GP Matrix And The Relationship Between Partial Orders In The Matrix

The special matrix and the relationship between the partial order of matrix has always been a hot issue in the research of matrix theory.The GP matrix is a special type of matrix,which occupies an important position in the special matrix.At the same time,the relationship between the GP matrix and other special matrices also lays a certain foundation for studying the partial order of the matrix.Partial ordering of matrix is an important research direction in matrix theory,it is closely related to other disciplines such as statistical multivariate analysis,cybernetics,and quantum mechanics,laying a solid foundation for the further development of other disciplines.In matrix partial order,studying the relationship between various partial orders is helpful to improve the theory of matrix partial order,so it is necessary to delve into the relationship between partial orders.Based on the research of experts and scholars,this thesis further gives the relevant properties of the GP matrix,obtains the relationship between the GP matrix and other important special matrices,and links the discussion of the GP matrix to the matrix partial order.Subsequently,the relationship between the partial ordering of the matrix was studied in depth using matrix decomposition and special matrices,and some new conclusions were obtained.The content of this article is divided into three parts:The first chapter gives the development and results of GP matrices and matrix partial ordering,as well as the definitions and lemmas to be used in this article.Such as:Definitions of various generalized inverses of matrices,fourteen special matrices,and five major partial orders,etc.The core-nilpotent decomposition of the matrix,the Hartwig-Spindelb?ck decomposition,and the equivalent description of the five partial orders,etc.The second chapter studies the problems related to GP matrix.First of all,the equivalent description of GP matrix is discussed in detail,and a more comprehensive description of GP matrix is obtained.Secondly,the relationship between GP matrix and other special matrices is studied by the nature of GP matrix,which makes the relationship between them more clear.Finally,the decomposition of the five matrix partial order under the condition of GP matrix obtained by the matrix Hartwig-Spindelbock decomposition,the GP matrix this special matrix is linked to the matrix partial order,enriching the content of matrix partial order.The third chapter mainly discusses the relationship between the five partial order.On the minus order and the star order;minus order and sharp order;star order and sharp order;core order and minus,star,sharp order;The relationship between the Lowner order and minus,star,sharp,and core order was systematically studied,and the clear correlation between these five partial orders was given from the concrete examples,as well as the addition of some special conditions after the association.Thus,the intrinsic relation of these five important partial order is obtained,and the matrix partial order theory is further improved.