Perturbation analysis of a matrix differential equation $\dot x=ABx$

TL;DR

This paper investigates the first order perturbations of matrix pairs under contragredient equivalence, identifying canonical pairs where these perturbations are always nonzero, with implications for matrix differential equations.

Contribution

It characterizes all canonical matrix pairs for which the first order induced perturbations are nonzero under contragredient equivalence.

Findings

Identifies conditions for nonzero first order perturbations

Provides a classification of canonical pairs in this context

Links perturbation analysis to matrix differential equations

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. Garc\'{\i}a-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 $(A + \widetilde A, B+\widetilde B)$ close to $(A,B)$ can be reduced by contragredient equivalence transformations that smoothly depend on the entries of $\widetilde A$ and $ \widetilde B$. Each perturbation $(\widetilde A,\widetilde B)$ of $(A,B)$ defines the first order induced perturbation $A\widetilde{B}+\widetilde{A}B$ of the matrix $AB$, which is the first order summand in the product $(A +\widetilde{A})(B+\widetilde{B}) = AB + A\widetilde{B}+\widetilde{A}B+ \widetilde A \widetilde B$. We find all canonical matrix pairs $(A,B)$, for which the first order induced perturbations $A\widetilde{B}+\widetilde{A}B$ are nonzero for all nonzero perturbations in the normal form of Garc\'{\i}a-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations $\dot x=Cx$, 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 $(S^{-1}AR,R^{-1}BS)$.