抄録
This paper considers the stability conditions of sign matrices and the minimal structures of stable matrices with diagonal sign patterns (0, -, …, -), (0, 0, -, …, -) and (0, 0, 0, -, …, -). Sign matrix (sign pattern (+, -, 0) of each element of a matrix) represents the interconnection structure of a system and so this paper deals with the structural stability of linear systems. Stability of a matrix A is closely related to the digraph associated with A and the diagonal sign pattern of A. Stable matrices with minimal number of nonzero elements are fundamental for stability consideration of sign matrices. Digraphs associated with sign matrices are assumed to be strongly connected. If associated digraph is a simple loop, the matrix can be stable iff at least n-1 diagonal elements are negative. For (0, -, …, -) type n×n matrices it is found minimal structure is unique and is a simple loop with n edges. For (0, 0, -, …, -) and (0, 0, 0, -, …, -) type matrices there are many minimal structures that contain n+1 edges. Some results of computer classification and stability check done for sign matrices with diagonal sign patterns (0, 0, 0, -), (0, -, -, -), (0, 0, -, -) and (0, 0, 0, -, -) are also included in this paper.
