Normal forms of "near similarity" transformations and linear matrix equations

    Authors

    A. Tovbis

    Comments

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    Abbreviated Journal Title

    Linear Alg. Appl.

    Keywords

    similarity transformations; normal forms; linear matrix equations; functional equations; discretized and singularly perturbed matrix; equations; SINGULAR DIFFERENTIAL-OPERATORS; Mathematics, Applied; Mathematics

    Abstract

    A formal solution to a linear matrix differential equation with irregular singularity t(1-r)Y'(t) = A(t)Y(t), where r is an element of Z(+) and the matrix-valued function A(t) is analytic at t = infinity, was obtained via reduction of the coefficient A(t) to its Jordan form. The same approach was also utilized to find formal solutions to difference equations and to singularly perturbed differential equations. The linear change of variables Y = TX, where X is the new unknown matrix, generates the transformation A -- > T(-1)AT - t(1-r)T(-1)T'. When r > 0, this transformation can be considered as a "small perturbation" of the similarity transformation A -- > T(-1)AT. Various normal forms of these two transformations could be found in the literature. The emphasis of the present paper is to describe some classes of "near similarity" transformations that have the same normal forms as A -- > T(-1)AT. Obtained results are used to construct formal solutions to matrix functional equations and to discretized differential equations. (C) 2000 Elsevier Science Inc. All rights reserved. AMS classification: 15A; 34E; 39A; 39B.

    Journal Title

    Linear Algebra and Its Applications

    Volume

    317

    Issue/Number

    1-3

    Publication Date

    1-1-2000

    Document Type

    Article

    Language

    English

    First Page

    13

    Last Page

    40

    WOS Identifier

    WOS:000089460700002

    ISSN

    0024-3795

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