<abstract xmlns="http://www.w3.org/1999/xhtml">Two complex matrix pairs (<italic>A</italic>, <italic>B</italic>) and (<italic>A′</italic>, <italic>B′</italic>) are contragrediently equivalent if there are nonsingular <italic>S</italic> and <italic>R</italic> such that (<italic>A′</italic>, <italic>B′</italic>) = (<italic>S</italic>−1<italic>AR</italic>, <italic>R</italic>−1<italic>BS</italic>). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (<italic>A</italic>, <italic>B</italic>) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (<italic>A</italic> + <italic>A͠</italic>, <italic>B</italic> + <italic>B͠</italic>) close to (<italic>A</italic>, <italic>B</italic>) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of <italic>A͠</italic> and <italic>B͠</italic>. Each perturbation (<italic>A͠</italic>, <italic>B͠</italic>) of (<italic>A</italic>, <italic>B</italic>) defines the first order induced perturbation <italic>AB͠</italic> + <italic>A͠B</italic> of the matrix <italic>AB</italic>, which is the first order summand in the product (<italic>A</italic> + <italic>A͠</italic>)(<italic>B</italic> + <italic>B͠</italic>) = <italic>AB</italic> + <italic>AB͠</italic> + <italic>A͠B</italic> + <italic>A͠B͠</italic>. We find all canonical matrix pairs (<italic>A</italic>, <italic>B</italic>), for which the first order induced perturbations <italic>AB͠</italic> + <italic>A͠B</italic> 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 <italic>ẋ</italic> = <italic>Cx</italic>, whose product of two matrices: <italic>C</italic> = <italic>AB</italic>; using the substitution <italic>x</italic> = <italic>Sy</italic>, one can reduce <italic>C</italic> by similarity transformations <italic>S</italic>−1<italic>CS</italic> and (<italic>A</italic>, <italic>B</italic>) by contragredient equivalence transformations (<italic>S</italic>−1<italic>AR</italic>, <italic>R</italic>−1<italic>BS</italic>).</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 + 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−1CS and (A, B) by contragredient equivalence transformations (S−1AR, R−1BS).

Perturbation analysis of a matrix differential equation <italic>ẋ</italic> = <italic>ABx</italic>

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Perturbation analysis of a matrix differential equation  = 

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....

Perturbation analysis of a matrix differential equation ẋ = ABx

Published Online: Oct 03, 2018

Page range: 97 - 104

Received: Nov 09, 2017

Accepted: Apr 02, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00007

Keywords
Contragredient equivalence, Miniversal deformation, Perturbation

© 2018 M. Isabel García-Planas and Tetiana Klymchuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Perturbation analysis of a matrix differential equation ẋ = ABx

Published Online: Oct 03, 2018

Page range: 97 - 104

Received: Nov 09, 2017

Accepted: Apr 02, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00007

KeywordsContragredient equivalence, Miniversal deformation, Perturbation

© 2018 M. Isabel García-Planas and Tetiana Klymchuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Contragredient equivalence, Miniversal deformation, Perturbation