## A Generalization
of ‘PolynomialExtendedGCD’
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.

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