On the closed representation for the inverses of Hessenberg matrices

Authors:
J. Abderramán Marrero;V. Tomeo
Affiliations:
Department of Mathematics Applied to Information Technologies, Telecommunication Engineering School, (ETSIT-U.P.M.) Madrid Tech. University, Avda Complutense s/n. Ciudad Universitaria, 28040 Madri ...;Department of Algebra, School of Statistics, (E.U.E.-U.C.M.) University Complutense, Avda de Puerta de Hierro s/n. Ciudad Universitaria, 28040 Madrid, Spain
Venue:
Journal of Computational and Applied Mathematics
Year:
2012

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Abstract

The general representation for the elements of the inverse of any Hessenberg matrix of finite order is here extended to the reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order, and some elementary properties of the inverse. These results are applied on the resolvent matrix associated to a finite Hessenberg matrix in standard form. Two examples on the unit disk are given.