An Implicit Q Theorem for Hessenberg-like Matrices

The implicit Q theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of, for example, implicit QR algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagonal matrices, proving thereby also in the symmetric case its power. Currently there is a growing interest to so-called semiseparable matrices. These matrices can be considered as the inverses of tridiagonal matrices. In a similar way, one can consider Hessenberg-like matrices as the inverses of Hessenberg matrices. In this paper, we formulate and prove an implicit Q theorem for the class of Hessenberg-like matrices. We introduce the notion of strongly unreduced Hessenberg-like matrices and also a method for transforming matrices via orthogonal transformations to this form is proposed. Moreover, as the theorem is valid for Hessenberg-like matrices it is also valid for symmetric semiseparable matrices. The research was partially supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). This research was partially supported by by MIUR, grant number 2004015437 (third author). The scientific responsibility rests with the authors.