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

Two complex matrix pairs $(A,B)$ and $(A^{\mathrm{\prime }},B^{\mathrm{\prime }})$ are contragrediently equivalent if there are nonsingular $S$ and $R$ such that $(A^{\mathrm{\prime }},B^{\mathrm{\prime }})=(S^{-1}AR,R^{-1}BS)$. 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 $(A+\tilde{A},B+\tilde{B})$ 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 $(\widetilde A,\widetilde B)$of$(A,B)$definesthefirstorderinducedperturbation$A\widetilde{B}+\widetilde{A}B$ofthematrix$AB$, 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 $A\tilde{B}+\tilde{A}B$ 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 $(S^{-1}AR,R^{-1}BS)$.