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Construction of unitary and normal companion matrices. (English)
Linear Algebra Appl. 202, 193-220 (1994).
For a polynomial \$f(z)\$ with unimodular zeros, two methods are presented to construct a unitary matrix whose characteristic polynomial is \$f(z)\$. It seems that the problem in question is close to the unitary eigenvalue problem, i.e. finding eigenvalues of a given unitary matrix. Any unitary matrix can be reduced to a unitary Hessenberg form by orthogonal transformations and the unitary Hessenberg form can be characterized by \$n\$ parameters \$γ\sb k\$, \$k = 1,2,\dots,n,\$ which are called reflection coefficients or Schur parameters. There are two sequences of polynomials associated with the unitary Hessenberg form, one is called Szegö polynomials and the other Sturm sequence of polynomials. These two sequences for polynomials lead to two methods for constructing the unitary Hessenberg form whose characteristic polynomial is given. Of interest is to investigate the relationships between the authors’ methods and these unitary Hessenberg methods.
Reviewer: C.He (Chemnitz)