Research On The Matrix Of Arbitrary Symbol Patterns

As an important part of combinatorial mathematics, the research of sign pattern matrix’s qualitative theory plays an important role. This paper mainly related to the knowledge of sign pattern matrix and using the Nilpotent-Jacobian method to prove the sign pattern matrix,complex sign pattern matrix and ray pattern matrix are spectrally arbitrary and minimally spectrally arbitrary.At the beginning of the passage, the research background, significance and related concepts of sign pattern and combinatorial mathematics are introduced. And its research status and process are introduced.Next, the main research contents of this thesis are given, it including the following parts :In the first part, the spectrally arbitrary and minimally spectrally arbitrary of a class of sign pattern matrix is studied by using the Nilpotent-Jacobian method.In the second part, a study on the spectrally arbitrary and minimal spectrally arbitrary of three complex sign patterns.In the third part, a class of ray pattern matrix is given, and using the Nilpotent-Jacobian method of ray pattern to prove the ray pattern matrix is spectrally arbitrary.