On the eigenvectors of Toeplitz matrices

Although every nonzero vector x ∈ Cn can be an eigenvector of a nonscalar Toeplitz matrix T, this assertion is generally false if symmetry is additionally required of T. It is shown that every symmetric or skew-symmetric vector is an eigenvector of a symmetric Toeplitz (nonscalar) matrix. A problem in matrix analysis that results in the need to characterize such eigenvectors is described.