Greatest common divisor of two polynomials

Abstract If a(λ) and b(λ) are two polynomials and A is the companion matrix of a(λ), then the matrix b(A) has rank δ(a)− k, where kis the degree of the greatest common divisor of a(λ) and b(λ). It is shown that, if the first k columns of b(A) are expressed as linear combinations of the remaining δ(a) − k columns (which are linearly independent), then the greatest common divisor is given by the coefficients of column k + 1 in these expressions.