On the eigenvectors of symmetric Toeplitz matrices

This paper presents a number of results concerning the eigenvectors of a symmetric Toeplitz matrix and the location of the zeros of the filters (eigenfilters) whose coefficients are the elements of the eigenvectors. One of the results is that the eigenfilters corresponding to the maximum and minimum eigenvalues, if distinct, have their zeros on the unit circle, while the zeros of the other eigenfilters may or may not have their zeros on the unit circle. Even if the zeros of the eigenfilters of a matrix are all on the unit circle, the matrix need not be Toeplitz. Examples are given to illustrate the different properties.