### 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. 2000 Mathematics Subject Classification 58K40, 58K50, 58K60, 32S25.

Original language | English |
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Pages (from-to) | 738-762 |

Journal | Proceedings of the London Mathematical Society |

Volume | 89 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2004 |

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## Cite this

Bruce, J. W., & Tari, F. (2004). On Families of Square Matrices.

*Proceedings of the London Mathematical Society*,*89*(3), 738-762. https://doi.org/10.1112/S0024611504014911