Spectral Characterizations for Hermitian Centrosymmetric K-Matrices and Hermitian Skew-Centrosymmetric K-Matrices

Abstract

Let A and K be real symmetric matrices with K2 =I. In the article "A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices" [D. Tao and M. Yasuda, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 885--895], it was shown that (1) AK=KA if and only if the spectrum of A equals the spectrum of KA up to sign and (2) AK=-KA if and only if the spectrum of A equals the spectrum of KA multiplied by i. This paper extends these spectral characterizations from the case of real symmetric matrices to that of self-adjoint compact linear operators in a complex Hilbert space. Some consequences of these results are mentioned, including an application that describes the correspondence between the spectrum of a real symmetric Toeplitz matrix T and its associated Hankel matrix JT, where J is the so-called exchange matrix.