A characterization of Toeplitz and Hankel circulants

A number of propositions of the following type is proved: A Toeplitz matrix $T$ is a circulant if and only if $T$ has an eigenvector $e$ with all the components equal to one. These propositions characterize the circulants (and, more generally, the $\phi$-circulants), as well as their Hankel counterparts, in the sets of all Toeplitz and Hankel matrices, respectively. Bibl. 2 titles.