Direct and Inverse Unitary Eigenproblems in Signal Processing: An Overview

Abstract

Let H n denote the set of unitary upper Hessenberg matrices of order n with nonnegative sub diagonal elements. These matrices bear many similarities with real symmetric tridiagonal matrices, both in terms of their structure, their underlying connections with orthogonal polynomials, and the existence of efficient algorithms for solving eigenroblems for these matrices. The Schur parameterization of H n provides the means for the development of efficient algorithms for finding eigenvalues and eigenvectors of these matrices. These algorithms include the Q R algorithm for unitary Hessenberg matrices [9], an algorithm for solving the orthogonal eigenproblem using two half-size singular value decompositions [1], a divide-and-conquer method [10, 5], an approach based on matrix pencils [6], and a unitary analog of the Sturm sequence method [7]. Aspects of inverse eigenproblems for unitary Hessenberg matrices are considered in [3] and efficient algorithms for constructing a unitary Hessenberg matrix from spectral data are presented in [13, 3].