 
Summary: Constructing a Unitary Hessenberg Matrix from Spectral Data
Gregory Ammar 1 , William Gragg 2 , Lothar Reichel 3
In memory of Peter Henrici
Abstract
We consider the numerical construction of a unitary Hessenberg matrix from spectral data using
an inverse QR algorithm. Any unitary upper Hessenberg matrix H with nonnegative subdiagonal
elements can be represented by 2n \Gamma 1 real parameters. This representation, which we refer to as
the Schur parameterization 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 rankone unitary perturbation of H: The
eigenvalues of the perturbation strictly interlace the eigenvalues of H on the unit circle.
AMS(MOS) Subject Classification: 15A18, 65F15.
Keywords: inverse eigenvalue problem, unitary matrix, orthogonal polynomial.
Presented at the NATO Advanced Study Institute on Numerical Linear Algebra, Digital Signal
Processing, and Parallel Algorithms, Leuven, Belgium, August, 1988. This research was supported
in part by the National Science Foundation under grant DMS8704196, and by the Foundation
Research Program of the Naval Postgraduate School.
This paper appears in: Numerical Linear Algebra, Digital Signal Processing and Parallel Algo
rithms, G.H. Golub and P. Van Dooren, eds., NATO ASI Series, Vol. F70, SpringerVerlag, Berlin,
1991, pp. 385395.
