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

DOI10.21042/AMNS.2018.1.00007arXiv1808.06506WikidataQ115233668 ScholiaQ115233668MaRDI QIDQ6305605

María Isabel García-Planas, Tetiana Klymchuk

Publication date: 17 August 2018

Abstract: Two complex matrix pairs $(A,B)$ and $(A^{\prime },B^{\prime })$ are contragrediently equivalent if there are nonsingular $S$ and $R$ such that $(A^{\prime },B^{\prime })=(S^{-1}AR,R^{-1}BS)$. 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 $(A+widetildeA,B+widetildeB)$ close to $(A,B)$ can be reduced by contragredient equivalence transformations that smoothly depend on the entries of $widetildeA$ and $widetildeB$. Each perturbation $(widetildeA,widetildeB)$ of $(A,B)$ defines the first order induced perturbation $AwidetildeB+widetildeAB$ of the matrix $AB$, which is the first order summand in the product $(A+widetildeA)(B+widetildeB)=AB+AwidetildeB+widetildeAB+widetildeAwidetildeB$. We find all canonical matrix pairs $(A,B)$, for which the first order induced perturbations $AwidetildeB+widetildeAB$ 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 $dotx=Cx$, 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 $(S^{-1}AR,R^{-1}BS)$.

Mathematics Subject Classification ID

Linear transformations, semilinear transformations (15A04) Canonical forms, reductions, classification (15A21)

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