Fiedler companion linearizations for rectangular matrix polynomials

De Teran, Fernando and Dopico, Froilan and Mackey, D. Steven (2011) Fiedler companion linearizations for rectangular matrix polynomials. [MIMS Preprint]

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Abstract

The development of new classes of linearizations of square matrix polynomials that generalize the classical first and second Frobenius companion forms has attracted much attention in the last decade. Research in this area has two main goals: finding linearizations that retain whatever structure the original polynomial might possess, and improving properties that are essential for accurate numerical computation, such as eigenvalue condition numbers and backward errors. However, all recent progress on linearizations has been restricted to square matrix polynomials. Since rectangular polynomials arise in many applications, it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials. In this paper, the family of Fiedler linearizations is extended from square to rectangular matrix polynomials, and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials. Fiedler linearizations are one of the most important classes of linearizations introduced in recent years, but their generalization to rectangular polynomials is nontrivial, and requires a completely different approach to the one used in the square case. To the best of our knowledge, this is the first class of new linearizations that has been generalized to rectangular polynomials.

Item Type: MIMS Preprint
Uncontrolled Keywords: rectangular matrix polynomials, Fiedler pencils, linearizations, minimal indices, minimal bases
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr. D. Steven Mackey
Date Deposited: 13 Dec 2011
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1732

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