Perturbation analysis of a matrix differential equation ¿x=ABx

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-1AR,R-1BS). 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+˜A,B+˜B) close to (A,B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of ˜A and ˜B. Each perturbation (˜A,˜B) of (A,B) defines the first order induced perturbation A˜B+˜AB of the matrix AB, which is the first order summand in the product (A+˜A)(B+˜B)=AB+A˜B+˜AB+˜A˜B. We find all canonical matrix pairs (A,B), for which the first order induced perturbations A˜B+˜AB 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 ¿x=Cx, whose product of two matrices: C=AB; using the substitution x=Sy, one can reduce C by similarity transformations S-1CS and (A,B) by contragredient equivalence transformations (S-1AR,R-1BS)

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Postprint (author's final draft)