On the eigenstructure of Toeplitz matrices

In spite of the fact that Toeplitz eigenvalue problems have important applications in signal processing, relatively little is known about the eigenstructure of finite real symmetric Toeplitz matrices. In this paper, we exhibit two facts about the Toeplitz eigenstructure problem. First we show that any reciprocal or anti-reciprocal n vector is an eigenvector for at least an [(n + 1)/2]-dimensional linear space of real symmetric n × n Toeplitz matrices. The second result is the fact that every n × n Toeplitz matrix can be imbedded into an (n + 1) × (n + 1) Toeplitz matrix which has a repeated smallest eigenvalue.