一个新的极小谱任意复符号模式矩阵

A New Matrix of Minimally Spectrally Arbitrary Complex Sign Patterns

若给定任意一个n阶首1复系数多项式f(λ),都存在一个复矩阵B∈Q(A),使得的特征多项式为f(λ),则称n×n复符号模式矩阵A是谱任意的.如果A是一个谱任意复符号模式矩阵且A的任意真子模式都不是谱任意的,那么A是一个极小谱任意复符号模式矩阵.本文扩展了N-J方法证明了一个的复符号模式矩阵是极小谱任意的n≥4.

A complex sign pattern matrix A of order n is a spectrally arbitrary pattern if given any monic polynomial f（λ） with coefficients from C of order n,there exists a complex matrix B in Q（A） such that the characteristic polynomial of B is f（λ）.If A is a spectrally arbitrary complex sign pattern matrix,and no proper subpattern of A is spectrally arbitrary,then A is a minimal spectrally arbitrary complex sign pattern matrix.In this paper,we extend the Nilpotent-Jacobian method to prove a complex sign pattern matrix is minimally spectrally arbitrary pattern for all orders n≥4.