Revisiting the matrix polynomial greatest common divisor

Abstract

In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices P i (λ) ∈ F[λ] m i ×n , i = 1, . . . , k with coefficients in a generic field F, and with common column dimension n. We give necessary and sufficient conditions for a matrix G(λ) ∈ F[λ] ℓ×n to be a GCRD using the Smith normal form of the m × n compound matrix P (λ) obtained by concatenating P i (λ) vertically, where m = k i=1 m i . We also describe the complete set of degrees of freedom for the solution G(λ), and we link it to the Smith form and Hermite form of P (λ). We then give an algorithm for constructing a particular minimum rank solution for this problem when F = C or R, using state-space techniques. This new method works directly on the coefficient matrices of P (λ), using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of P (λ).