On the eigenvectors of Toeplitz matrices
Although every nonzero vector $x \in {\bf C}^n$ can be an eigenvector of a nonscalar Toeplitz matrix $T$, an analogous assertion is in general false if one additionally requires that $T$ be a symmetric matrix. 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 leads to the necessity of characterizing such eigenvectors is described.