H-(反)对称矩阵的广义特征值反问题

摘要

讨论H矩阵的性质,给出H-对称矩阵和H-反对称矩阵的结构,证明若x是H-对称矩阵或H-反对称矩阵A -λB的特征向量,则x是H-对称向量或H-反对称向量,或者x可以由H-对称向量及H-反对称向量线性表示,并根据A-λB的特征向量的上述特点,得到H-对称矩阵和H-反对称矩阵的广义特征值反问题AX=BXA解的表达式.%Properties of H-matrices were discussed, the structures of H-symmetric and H-antisymmetric matrices were given, and it was proven that when x was an eigenvector of H-symmetric matrices or H-antisymmetric matrices A-λB, x would be either an H-symmetric vector, or H-antisymmetric vector, or x could be expressed by linear combination of H-symmetric vector with H-antisymmetric vector. Based on a-bove-mentioned feature of eigenvector of A-λB, the expression of solution to inverse problem AX=BXA of generalized eigenvalue of H-symmetric matrices and H-antisymmetric matrices were obtained.