Some Spectral Properties of Hermitian Toeplitz Matrices
William F. Trench · SIAM Journal on Matrix Analysis and Applications · 1994
Necessary conditions are given for the Hermitian Toeplitz matrix $T_n = ( t_{r - s} )_{r,s = 1}^n $ to have a repeated eigenvalue $\lambda $ with multiplicity $m > 1$ and for an eigenpolynomial of $T_n $ associated with $\lambda $ to have a given number of zeros off the unit circle $| z | = 1$. It is assumed that $t_r = \frac{1} {{2\pi }}\int_{ - \pi }^\pi {f( \theta )} e^{ - ir\theta } d\theta\, ( 0 \leq r \leq n - 1)$, where f is real-valued and in $L( - \pi ,\pi )$. The conditions are given in terms of the number of changes in sign of $f( \theta ) - \lambda $.