Constructing a Unitary Hessenberg Matrix from Spectral Data

Abstract

We consider the numerical construction of a unitary Hessenberg matrix from spectral data using an inverse QR algorithm. Any upper unitary Hessenberg matrix H with nonnegative subdiagonal elements can be represented by 2n − 1 real parameters. This representation, which we refer to as the Schur parametrization of H, facilitates the development of efficient algorithms for this class of matrices. We show that a unitary upper Hessenberg matrix H with positive subdiagonal elements is determined by its eigenvalues and the eigenvalues of a rank-one unitary perturbation of H. The eigenvalues of the perturbation strictly interlace the eigenvalues of H on the unit circle.