Möbius Transformations of Matrix Polynomials

Mackey, D. Steven and Mackey, Niloufer and Mehl, Christian and Mehrmann, Volker (2014) Möbius Transformations of Matrix Polynomials. [MIMS Preprint]

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Abstract

We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated.

Item Type: MIMS Preprint
Uncontrolled Keywords: Mobius transformation, generalized Cayley transform, matrix polynomial, matrix pencil, Smith form, local Smith form, elementary divisors, partial multiplicity sequence, Jordan characteristic, Jordan chain, invariant pair, compound matrices, minimal indices, minimal bases, structured linearization, palindromic matrix polynomial, alternating matrix polynomial.
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: 07 May 2014
Last Modified: 20 Oct 2017 14:13
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2131

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