Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices Original Research Article

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 λ1 greater-or-equal, slanted λ2 greater-or-equal, slanted cdots, three dots, centered greater-or-equal, slanted λn is the spectrum of real symmetric centrosymmetric matrices S1 and S2 such that for S1 an eigenvector corresponding to λ1 is the all ones vector and for S2 an eigenvector corresponding to λ1 is the vector with components 1 and −1 alternately. The proof is constructive. Then, we derive an improved condition on image such that S1 = λ1E, 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.