Abstract

We derive a readily computable sufficient condition for the existence of a nonnegative symmetric circulant matrix having a prescribed spectrum. Moreover, we prove that any set ?1??2 ?? ??nis the spectrum of real symmetric centrosymmetric matrices S1 and S2 such that for S1 an eigenvector corresponding to ?1 is the all ones vector and for S2 an eigenvector corresponding to ?i is the vector with components 1 and -1 alternately. The proof is constructive. Then, we derive an improved condition on {?k}k=1n such that S1 = ?1 E, where E is a stochastic symmetric centrosymmetric matrix. Finally, we propose an algorithm to compute the eigenvalues of some real symmetric centrosymmetric matrices. All the numerical procedures are based on the use of the Fast Fourier Transform. © 2004 Elsevier Inc. All rights reserved.