Spectral Evolution of a One-Parameter Extension of a Real Symmetric Toeplitz Matrix

Let $T_n = (t_{i - j} )_{i,j = 1}^n (n\geqq 3)$ be a real symmetric Toeplitz matrix such that $T_{n - 1} $ and $T_{n - 2} $ have no eigenvalues in common. The evolution of the spectrum of $T_n $ as the parameter $t = t_{n - 1} $ varies over $( - \infty ,\infty )$ is considered. It is shown that the eigenvalues of $T_n $ associated with symmetric (reciprocal) eigenvectors are strictly increasing functions of t, while those associated with the skew-symmetric (anti-reciprocal) eigenvectors are strictly decreasing. Results are obtained on the asymptotic behavior of the eigenvalues and eigenvectors at $t \to \pm \infty $, and on the possible orderings of eigenvalues associated with symmetric and skew-symmetric eigenvectors.