The Minimum Completions And Covers Of Symmetric,Hankel Symmetric And Centrosymmetric Doubly Substochastic Matrices

In this paper,we mainly discuss and describe the minimum completions and covers of symmetric,Hakel symmetric and centrosymmetric doubly substochastic matrices based on the definitions and properties of doubly stochastic matrices,dou-bly substochastic matrices and sub-defect.The minimum completion of a doubly substochastic matrix is a doubly stochastic matrix with the smallest order that contains the doubly substochastic matrix as a submatrix;the cover of a doubly substochastic matrix is a doubly stochastic matrix of the same order,in which the elements at each position are greater than or equal to the elements in the same position.In Chapter 1,we mainly introduce the background knowledge,as well as the definitions and theorems used later.In Chapter 2,we study the minimum doubly stochastic completions of sym-metric,Hankel symmetric,and centrosymmetric doubly substochastic matrices that maintain the corresponding symmetrical structure.And we extend the definition of sub-defect from square matrices to non-square matrices.The main results are as follows:A symmetric doubly substochastic matrix of order n.If its sub-defect is k,then there exists a symmetric doubly stochastic matrix of order n+k containing it as a submatrix.Similarly,we have shown that when we replace "symmetric" with“Hankel symmetric”,the same is true.A centrosymmetric doubly substochastic matrix of order n.If its sub-defect is k,then when k is even,there exists a centrosymmetric doubly stochastic matrix of order n+k containing it as a submatrix;when k is odd,there exists a centrosym-metric doubly stochastic matrix of order n+k+1 containing it as a submatrix.Similarly,we have shown that when we replace "centrosymmetric" with "symmetric and Hankel symmetric”,the same is true.In Chapter 3,we study how to change a doubly substochastic matrix into a doubly stochastic matrix of the same order while maintaining the original sym-metrical structure.The resulting matrix is called the cover of the original doubly substochastic matrix.