On a Sturm Sequence of Polynomials for Unitary Hessenberg Matrices

Abstract

Unitary matrices have a rich mathematical structure that is closely analogous to real symmetric matrices. For real symmetric matrices this structure can be exploited to develop very efficient numerical algorithms and for some of these algorithms unitary analogues are known. Here we present a unitary analogue of the bisection method for symmetric tridiagonal matrices. Recently Delsarte and Genin introduced a sequence of so-called $\gamma_n$-symmetric polynomials that can be used to replace the classical Szego polynomials in several signal processing problems. These polynomials satisfy a three-term recurrence relation and their roots interlace on the unit circle. Here we explain this sequence of polynomials in matrix terms. For an $n\times n$ unitary Hessenberg matrix, we introduce, motivated by the Cayley transformation, a sequence of modified unitary submatrices. The characteristic polynomials of the modified unitary submatrices $p_k(z), k=1,2,\ldots,n$ are exactly the $\gamma_n$-symmetric polynomials up to a constant. These polynomials can be considered as a sort of Sturm sequence and can serve as a basis for a bisection method for computing the eigenvalues of the unitary Hessenberg matrix. The Sturm sequence properties allow identification of the number of roots of $p_n(z)$, the characteristic polynomial of the unitary Hessenberg matrix itself, on any arc of the unit circle by computing the sign agreements of certain related real polynomials at a given point.