## 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*).