On Determinantal Regions Of Complex Matrices

Signed matrix theory is a new branch of combinatorial ma-trix theory, which involves the study of the properties of matricesthat depend only on the sign pattern of the matrices. The sub-ject, started by the Nobelist P.Samuelson who is an economist, isusually considered to have originated with the discussion on "qual-itative properties" of some problems in economics (see [16]). Be-cause of its great importance in economics, it has been paid wideattention by many researchers including economists, mathemati-cians and theoretical computer scientists. In 1995, R.A.Brualdi andB.L.Shader's book《Matrices of sign-solvable linear systems》([5])was published, which is the first monograph on signed matrix the-ory. In this book, they systematically summarized the results onthe subject and gave many new results. It makes signed matrixtheory being an active area of combinatorics.Recently, the research of signed matrix theory transfers fromreal field to complex field. In 1997, J.J.McDonald etc. generalizedthe concept of the qualitative class to ray pattern class, and stud-ied questions on the nonsingularity and determinantal regions of raypattern matrices. In 1998, Eschenbach etc. extended the conceptof the qualitative class in another way in [6], so called the complexsign pattern. In both ways of extension, a fundamental questionis out there: what is the characterization of nonsingular matricesin ray pattern or in complex sign pattern? In order to investigatethis problem, the definition of determinantal regions of matrices isreferred in [14]. In 2005, J.Y.Shao and H.Y.Shan did a deep re-search in determinantal regions of matrices in [20]. Some necessaryconditions of determinantal regions is given and all possible regionswere listed (22 types altogether 74 possible regions, 6 types were leftundetermined). Also an important parameter denoted by n_R(A) isreferred, and some questions were asked.Question 1: Is n_R(A) always finite when A ranges over all com-plex square matrices?Question 2: If the answer to Question 1 is true, determine themaximal possible value of n_R(A), and characterize cases with dif-ferent n_R(A). Similar questions may be raised for the complex sign pattern.We mainly discussed the parameter n_R(A) in this paper. Firsta kind of distance between two matrices with the same size is de-fined here. Then it is proved that the maximal possible value ofn_R(A) is 2, further more the characterization of case n_R(A)=2is given. After that all possible regions as determinantal regionsof ray pattern matrices are listed, there are 9 types and only onetype is left undetermined. The proof is also effective for complexsign pattern. So use these conclusions, 4 of the 6 types left in[20] are excluded for determinantal regions of complex sign patternmatrices.