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

Abstract

Two complex matrix pairs $(A,B)$ and $(A^{\prime },B^{\prime })$ are contragrediently equivalent if there are nonsingular $S$ and $R$ such that $(A^{\prime },B^{\prime })=(S^{-1}AR,R^{-1}BS)$. M.I. Garc\'ıa-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 $(A+\tilde{A},B+\tilde{B})$ close to $(A,B)$ can be reduced by contragredient equivalence transformations that smoothly depend on the entries of $\tilde{A}$ and $\tilde{B}$. Each perturbation $(\tilde{A},\tilde{B})$ of $(A,B)$ defines the first order induced perturbation $A\tilde{B}+\tilde{A}B$ of the matrix $AB$, which is the first order summand in the product $(A+\tilde{A})(B+\tilde{B})=AB+A\tilde{B}+\tilde{A}B+\tilde{A}\tilde{B}$. We find all canonical matrix pairs $(A,B)$, for which the first order induced perturbations $A\tilde{B}+\tilde{A}B$ are nonzero for all nonzero perturbations in the normal form of Garc\'ıa-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 $(S^{-1}AR,R^{-1}BS)$.