Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices
The authors derive a useful sufficient condition for the existence of a non-negative symmetric circulant matrix having a prescribed spectrum. They, moreover, prove that any set \(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\) of real numbers is the spectrum of real symmetric centrosymmetric matrices \(S_1\) and \(S_2\) such that an eigenvector of \(S_1\) for the eigenvalue \(\lambda_1\) is the all-ones vector and an eigenvector of \(S_2\) for the eigenvalue \(\lambda_1\) is the vector consisting of the entry \((-1)^{i-1}\) in the \(i\)-th component. The authors also propose an algorithm to compute the eigenvalues of some real symmetric centrosymmetric matrices, based on fast Fourier transform.
scientific article; zbMATH DE number 2119136