Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils

TL;DR

This paper develops simple normal forms for families of real matrices and matrix pencils that depend smoothly on parameters, extending Arnold's work on complex matrices to real cases and matrix pencils.

Contribution

It constructs minimal normal forms for parameter-dependent real matrices and matrix pencils, simplifying previous normal forms by Galin and Edelman et al.

Findings

Normal forms for real matrices depending smoothly on parameters.

Normal forms for matrix pencils depending smoothly on parameters.

Simplification of existing normal forms for these families.

Abstract

V. I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a simple normal form for a family of complex n-by-n matrices that smoothly depend on parameters with respect to similarity transformations that smoothly depend on the same parameters. We construct analogous normal forms for a family of real matrices and a family of matrix pencils that smoothly depend on parameters, simplifying their normal forms by D. M. Galin [Uspehi Mat. Nauk 27 (1) (1972) 241-242] and by A. Edelman, E. Elmroth, B. Kagstrom [Siam J. Matrix Anal. Appl. 18 (3) (1997) 653-692].