On an inverse eigenvalue problem for unitary Hessenberg matrices (Q1805209)

The authors consider the inverse eigenvalue problem for unitary Hessenberg matrices. It is proved that if \(\{\lambda_k\}^n_{k = 1}\) and \(\{\mu_k\}^{n - 1}_{k = 0}\) are two sets of strictly interlacing points on the unit circle in the complex plane, then there exists a unique unitary Hessenberg matrix \(H = H (\gamma_1, \gamma_2, \ldots, \gamma_n)\) such that the spectrum of the matrix \(\lambda (H) = \{\lambda_k\}^n_{k = 1}\) and \(\lambda (H_{n - 1}') = \{\mu_k\}^{n - 1}_{k = 1}\), where \(H_{n - 1}' = H (\gamma_1, \gamma_2, \ldots, \gamma_{n - 2}, \rho_{n - 1})\) and \(\rho_{n - 1} = (\gamma_{n - 1} + \overline {\mu}_0 \gamma_n)/(1 + \overline {\mu}_0 \overline {\gamma}_{n - 1} \gamma_n)\). A procedure for constructing the matrix \(H (\gamma_1, \gamma_2, \ldots, \gamma_n)\) is presented.