An observation on certain spectral properties of Toeplitz matrices (Q1193077)

For the generalized eigenvalue problem \(T_ n x=\lambda S_ n x\), where \(T_ n\) and \(S_ n\) are given \(n\times n\) Hermitian matrices and the generalized eigenvalues \(\lambda_ 0,\dots,\lambda_{n-1}\) are distinct and arranged in increasing order, the eigenpolynomial relative to \(\lambda_ r\) (\(0<r<n-1\)) is defined as \(U_ r(z)=[1,z,\dots,z^{n- 1}]\vec U^{(r)}\), where \(\vec U^{(r)}\) is the generalized eigenvector associated with \(\lambda_ r\). If \(T_ n\) and \(S_ n\) are Toeplitz, then the zeros of \(U_ r(z)\) either occur in reciprocal pairs or lie on the unit circle. The authors consider the problem of determining the minimal number of zeros of \(U_ r(z)\) lying on the unit circle. Firstly a result of \textit{P. Delsarte} and \textit{Y. Genin} [Lect. Notes Control Inf. Sci. 58, 194- 213 (1984; Zbl 0559.15017)] on the lower bound for the number of zeros on the unit circle of the eigenpolynomial associated with the \(i\)-th eigenvalue of a general Toeplitz matrix \(T_ n\) is extended to the generalized eigenvalue problem. Then, the case of rational Toeplitz matrices is considered. They prove that if \(T_ n\) and \(S_ n\) are rational matrices generated by the rational functions \(T(z)\) and \(S(z)\), respectively, then the number of zeros of \(U(z)\) on the unit circle is at least \(n-2(M+N)+1\), where \(M\) and \(N\) are related to the degrees of the polynomial defining the rational functions \(T(z)\) and \(S(z)\). As particular cases, for \(T_ n\) and \(S_ n\) being both banded or both tridiagonal the corresponding conclusions are obtained at once.