Generic families of matrix pencils and their bifurcation diagrams

TL;DR

This paper extends Arnold's work on smooth generic families of matrices to matrix pencils, analyzing their bifurcation diagrams with respect to similarity transformations depending smoothly on parameters.

Contribution

It introduces a framework for studying bifurcation diagrams of matrix pencils, generalizing Arnold's results from matrices to pencils.

Findings

Extended Arnold's bifurcation analysis to matrix pencils

Developed smooth generic families for matrix pencils

Provided bifurcation diagrams for these families

Abstract

V. I. Arnold [Russian Math. Surveys 26, no. 2, 1971, 29-43] constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the entries of matrices and got bifurcation diagrams of such families with a small number of parameters. We extend these results to pencils of matrices.