Revisiting the matrix polynomial greatest common divisor
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In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices P_i(λ)∈[]^m_i× n, i=1,…,k with coefficients in a generic field , and with common column dimension n. We give necessary and sufficient conditions for a matrix G()∈[]^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=∑_i=1^k m_i. We also describe the complete set of degrees of freedom for the solution G(), and we link it to the Smith form and Hermite form of P(). We then give an algorithm for constructing a particular minimum rank solution for this problem when = or , using state-space techniques. This new method works directly on the coefficient matrices of P(), 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().
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