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On families of square matrices

Published online by Cambridge University Press:  05 November 2004

J. W. Bruce  and
F. Tari
J. W. Bruce
Affiliation:
Department of Pure Mathematics, The University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, United Kingdom Current address: Deputy Vice Chancellor, University of Hull, Cottingham Road, Hall, HU6 7RX, United Kingdom. E-mail: j.w.bruce@hull.ac.uk
F. Tari
Affiliation:
Departamento Matemática, ICMC/USP São Carlos, Caixa Postal 668, CEP 13560-970, São Carlos (SP), Brazil. E-mail: tari@icmc.usp.br Current address: Department of Mathematical Sciences, University of Durham, South Road, Durham, DH1 3LE, United Kingdom
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Abstract

In this paper we classify families of square matrices up to the following natural equivalence. Thinking of these families as germs of smooth mappings from a manifold to the space of square matrices, we allow arbitrary smooth changes of co-ordinates in the source and pre- and post- multiply our family of matrices by (generally distinct) families of invertible matrices, all dependent on the same variables. We obtain a list of all the corresponding simple mappings (that is, those that do not involve adjacent moduli). This is a non-linear generalisation of the classical notion of linear systems of matrices. We also make a start on an understanding of the associated geometry.

Keywords

pencils of matricessingularitiesdeterminacy58K4058K5058K6032S25
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 89 , Issue 3 , November 2004 , pp. 738 - 762
Copyright
2004 London Mathematical Society

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Footnotes

The research of the second author was partially supported by a CNPq grant.