Spectral properties of finite Toeplitz matrices

The paper contains an investigation of certain spectral properties of finite Hermitian Toeplitz matrices. Some classical results relative to a constant Toeplitz matrix C are first extended to the polynomial matrix λI-C. Next, Carathéodory's representation based on the smallest eigenvalue of C is generalized to the case of an arbitrary eigenvalue. The splitting of each eigenspace of a real symmetric Toeplitz matrix C into its reciprocal and antireciprocal subspaces is then characterized. New identities are derived relating the characteristic determinants of the reciprocal and antireciprocal components of the Toeplitz submatrices of C. A special attention is brought to the inverse eigenvalue problem for Toeplitz matrices and some examples are given.