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Applied Mathematics and Nonlinear Sciences

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Available online: April 02, 2018

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

M. Isabel García Planas, Tetiana Klymchuk
Volume 3 - Issue 1.     Year 2018.     Pages 97–104.

DOI: 10.21042/AMNS.2018.1.00007


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 $(A + \widetilde{A}, B + \widetilde{B})$ close to $(A,B)$ can be reduced by contragredient equivalence transformations that smoothly depend on the entries of $\widetilde{A}$ and $\widetilde{B}$. Each perturbation $(\widetilde{A}, \widetilde{B})$ of $(A,B)$ defines the first order induced perturbation $A\widetilde{B}+\widetilde{A}B$ of the matrix $AB$, 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 $A\widetilde{B}+\widetilde{A}B$ 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 $\dot{x}=Cx$, 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 $(S^{-1}AR,R^{-1}BS)$.

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Applied Mathematics, Nonlinear Sciences


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