Abstract: For every singular Hermitian matrix A, there exists a k-th unit matrix R such that A is k degree R-symmetric. Suppose 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 explicit 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.

Key words: singular value decomposition , characteristic polynomial, eigen-pair, generalized symmetric matrix