Eigenvalue Localization for Symmetric Positive Toeplitz Matrices

Given a real symmetric matrix, several inclusion and exclusion intervals containing its eigenvalues can be given. In this paper, for symmetric positive Toeplitz matrices, we provide an inclusion interval and, under an additional hypothesis, we also give two disjoint intervals contained in the previous one and containing all the eigenvalues. Examples are included, showing that these two intervals are necessary and that they can provide precise information on the localization of the eigenvalues. Sufficient conditions for positive definiteness are included. Necessary and sufficient conditions for the total positivity of symmetric positive Toeplitz matrices are presented. A characterization of symmetric totally positive circulant matrices is also obtained.