Perturbation analysis of a matrix differential equation ẋ = ABx

M Isabel García-Planas,Tetiana Klymchuk

doi:10.21042/amns.2018.1.00007

M Isabel García-Planas, Tetiana Klymchuk

Open Access

https://doi.org/10.21042/amns.2018.1.00007

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Abstract

Abstract Two complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S −1 AR, R −1 BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + A͠, B + B͠) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of A͠ and B͠. Each perturbation (A͠, B͠) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + A͠)(B + B͠) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S −1 CS and (A, B) by contragredient equivalence transformations (S −1 AR, R −1 BS).

Highlights

We study a matrix differential equation x = ABx, whose matrix is a product of an m × n complex matrix A and an n × m complex matrix B
He constructed a miniversal deformation of a Jordan matrix J; i.e., a simple normal form to which all matrices J + E close to J can be reduced by similarity transformations that smoothly depend on the entries of E
We describe all canonical matrix pairs (A, B) of the form (2), for which the first order induced perturbations AB + AB are nonzero for all miniversal perturbations (A, B) = 0 in the normal form defined in [4]

Summary

Introduction

A notion of miniversal deformation was introduced by Arnold [1, 2] He constructed a miniversal deformation of a Jordan matrix J; i.e., a simple normal form to which all matrices J + E close to J can be reduced by similarity transformations that smoothly depend on the entries of E. García-Planas and Sergeichuk [4] constructed a miniversal deformation of a canonical pair (2) for contragredient equivalence (1). We describe all canonical matrix pairs (A, B) of the form (2), for which the first order induced perturbations AB + AB are nonzero for all miniversal perturbations (A, B) = 0 in the normal form defined in [4].

Miniversal deformations of counter matrix pairs

Main theorem