Normal forms of “near similarity” transformations and linear matrix equations

Abstract

A formal solution to a linear matrix differential equation with irregular singularity t 1−rY ′(t)=A(t)Y(t) , where r∈ Z + and the matrix-valued function A(t) is analytic at t=∞, 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 −1AT−t 1−rT −1T ′ . When r>0, this transformation can be considered as a “small perturbation” of the similarity transformation A→T −1AT . 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 −1AT . Obtained results are used to construct formal solutions to matrix functional equations and to discretized differential equations.