Isolated Replacement Set, The Determinant Range Of Complex Matrices, And Fully Intends To S ~ * Array,

As a branch of combinatorial matrix theory, signed matrix theory mainlystudies a matrix, s qualitative properties which only depend on the matrix, s signpattern.The subject originated from the discussion about the sign-solvable linearsystems which was started in 1947 by P. Samuelson, a Nobelist and an economist.Since then, signed matrix theory has attracted much attention from economists, mathematicians and computer scientists due to its important economic applica-tions and the beautiful interplay it afforded among linear algebra, combinatoricsand theoretical computer science. In 1995, the first book on signed matrix theory-《Matrices of Sign-solvable Linear Systems》([9])by R.A.Brualdi and B.L.Shadersummarized systematically the previous research results and also gave many newresults, which made signed matrix theory an active research field in combinatorics.Recently, some scholars have done a job which aims at generalizing the resultsin signed matrix theory from real field to complex field. In 1997, J.J.McDonaldetal. introduced ray nonsingular matrices as a generalization of real nonsingularmatrices in[28]. During the course of studying the determinantal regions of raypattern matrices, they advanced the concept of isolated transversal set, and theyalso gave a characterization of an isolated transversal set whose size is at most3. In Chapter2, we point out that an isolated transversal set is equivalent toan isolated permutation set(§2.1), and give graph theoretical characterizationsof an isolated permutation set(§2.2). In addition, we have solved a problem onjustifying the linear independence of some permutation matrices as an applicationof the characterizations of an isolated permutation set(§2.3).In 1998, C.A.Eschenbach etal. generalized real sign pattern matrices to com-plex sign pattern matrices in [11], and put up with some important open problems, one of which is "Giving all the possible regions of complex sign pattern matrices".In [41], J.Y.Shao and H.Y.Shan listed some possible regions of complex sign pat-tern matrices, in addition to these regions, they conjectured that all of the restregions—(B1)-(B11) can't be a determinantal region of any complex sign patternmatrix. (B1)-(B3) and (B6)-(B7) have been excluded in [41]. In Chapter3, weexclude (B8)(§3.2), and (B4),(B9),(B10)(§3.3).Totally quasi S~* matrices are an important matrix class on which [9]did aspecial discussion. A characterization on zero patterns of totally quasi S~* matriceswas given in [9]. In Chapter4, we independently obtain a characterization of signpatterns of totally quasi S~* matrices(§4.1), and give the structure of totally quasiS~* matrices after defining totally quasi S~* signed bipartite graphs and studyingthe structure of totally quasi S~* signed bipartite graphs(§4.2).Nearly quasi S~* matrices are a class of matrices which has a close relation-ship with quasi S~* matrices. In Chapter5, we characterize nearly quasi S~* ma-trices(§5.1), and give all of nearly quasi S~* matrices after defining nearly quasi S~* bipartite digraphs and studying the structure of nearly quasi S~* bipartite di-graphs(§5.2).