Block triangular miniversal deformations of matrices and matrix pencils

TL;DR

This paper introduces block triangular miniversal normal forms for matrices, matrix pencils, and contragredient matrix pencils, extending prior work on normal forms by providing a new structural approach.

Contribution

It presents new block triangular miniversal normal forms for families of matrices and matrix pencils, expanding the existing theory of normal forms with a novel structural perspective.

Findings

Normal forms are block triangular.

Extension of Arnold's and Edelman et al.'s normal forms.

Provides a new framework for analyzing matrix families.

Abstract

For each square complex matrix, V. I. Arnold constructed a normal form with the minimal number of parameters to which a family of all matrices B that are close enough to this matrix can be reduced by similarity transformations that smoothly depend on the entries of B. Analogous normal forms were also constructed for families of complex matrix pencils by A. Edelman, E. Elmroth, and B. Kagstrom, and contragredient matrix pencils (i.e., of matrix pairs up to transformations (A,B)-->(S^{-1}AR,R^{-1}BS)) by M. I. Garcia-Planas and V. V. Sergeichuk. In this paper we give other normal forms for families of matrices, matrix pencils, and contragredient matrix pencils; our normal forms are block triangular.