A generalization of "PolynomialExtendedGCD" and perturbed Sylvester Equations

A generalization of ‘olynomialExtendedGCD’ nd perturbed Sylvester Equations

Shigeki Matsumoto

shigeki@konan-u.ac.jp
faculty of science
Konan University
8-9-1, Okamoto, Kobe, 658-8501
Japan

Abstract

We extend a Mathematica command PolynomialExtendedGCD to the case that the coefficients of polynomials contain indeterminates. By virtue of such an extension of the command, one can solve the Sylvester (matrix) equation AX – XB = C explicitly even in the case that the square m x m matrix A and n x n matrix B have common eigenvalues, where C is an m x n matrix, X an (unknown) m x n matrix.

The asymptotic behavior (t -> 0) of the solution of perturbed (non-degenarate) Sylvester Equation (A + t ) X - XB = C plays an essential role in our argument.