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M. Isabel García-Planas and Tetiana Klymchuk


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 + , B + ) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of and . Each perturbation (, ) 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 + )(B + ) = 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 = 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).