The Structure of the Inverse to the Sylvester Resultant Matrix

Details The Structure of the Inverse to the Sylvester Resultant Matrix The Structure of the Inverse to the Sylvester Resultant Matrix

2001-04-11

arxiv.org/abs/math/0104118v1

Linear Algebra and Its Applications 85:191-202 (1987)

Mathematics

Rings and Algebras

11 pages

Scientific paper

Given polynomials a(z) of degree m and b(z) of degree n, we represent the inverse to the Sylvester resultant matrix of a(z) and b(z), if this inverse exists, as a canonical sum of m+n dyadic matrices each of which is a rational function of zeros of a(z) and b(z). As a result, we obtain the polynomial solutions X(z) of degree n-1 and Y(z) of degree m-1 to the equation a(z)X(z)+b(z)Y(z)=c(z), where c(z) is a given polynomial of degree m+n-1, as follows: X(z) is a Lagrange interpolation polynomial for the function c(z)/a(z) over the set of zeros of b(z) and Y(z) is the one for the function c(z)/b(z) over the set of zeros of a(z).

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