Sign Pattern Matrices Nilpotency

This paper includes four chapters, and it mainly studies the character and construction of allowing nilpotent sign pattern matrix, meanwhile, it proves the existence of allowing nilpotent sign pattern matrix. This paper includes two aspects: (1) character and construction of allowing nilpotent sign pattern matrix of index at most 2 and at most 3; (2) character and construction of allowing nilpotent sign pattern matrix of index at most k. Chapter three is impotent part in this paper, it proves the existence of allowing nilpotent sign pattern matrix of index at most k, and gives the constructing method of allowing nilpotent sign pattern matrix with arbitrary index.In the first part (chapter two), the paper mainly introduces the research on allowing nilpotent sign patterns of index at most 2 and at most 3, gives the equal three conditions of normal allowing nilpotent, such as Theorem 2.3.1. Based on this, we study character of allowing nilpotent sign patterns of index at most 2 and at most 3, and then obtain five methods of constructing allowing nilpotent sign patterns of index of at most 2 and at most 3.In the second part (chapter three), we obtain some character of allowing nilpotentsign pattern of index at most k based on the result of index at most 2 and at most 3, such as the minimal rank of N_k. Furthermore, we obtain five methods of constructingallowing nilpotent sign patterns of index at most k and prove the correctness of constructing method. Among these methods, the second one has the constructing character, the third and fourth ones are the extent of the result in chapter two.