Inverses of generalized Hessenberg matrices

Abstract

Some procedures are proposed for well known as well as new factorizations of the inverses of nonsingular generalized Hessenberg matrices . Moreover, we take advantage in such decompositions of low-rank structures of Hessenberg-like matrices . This suite of decompositions includes various cases when the matrix is nonsingular and also the case when has a unique factorization. Finally as a main result, a simple and constructive characterization of the nonsingular generalized Hessenberg matrices and their inverses are presented.

Keywords:

• Hessenberg matrix
• inverse matrix
• matrix factorization
• structured matrix

AMS Subject Classifications:

• 15A09
• 15A23
• 15A33
• 65F05

Acknowledgements

The authors thank to the reviewers by their valuable help in the improvement of this article.