广义对称矩阵的特征问题及其奇异值分解

对于任意奇异的Hermitian矩阵A, 存在一个非平凡k次单位矩阵R使得A为k次R-对称矩阵。 给定k次单位矩阵R, 给出了k次R-对称矩阵的特征对的性质、特征多项式的计算公式和奇异值分解, 并利用此类广义对称矩阵的特殊结构将其特征问题降阶, 转化成若干个低价矩阵的特征问题来计算。

Eigen-problem and singular value decomposition of the generalized symmetric matrix

For every singular Hermitian matrix A, there exists a k-th unit matrix R such that A is k degree R-symmetric. Sup- pose that a k- th unit matrix R is given for a k degree R-symmetric matrix A. It presents the properties of its eigen-pairs, the ex- plicit expression of its characteristic polynomial, and its singular value decomposition. Moreover, an eigen-problem of A can be reduced to multi eigen-problems of matrices of small dimensions by applying its structure.