Revisiting the matrix polynomial greatest common divisor

In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices $P_i(\lambda)\in \F[\la]^{m_i\times n}$, $i=1,\ldots,k$ with coefficients in a generic field $\F$, and with common column dimension $n$. We give necessary and sufficient conditions for a matrix $G(\la)\in \F[\la]^{\ell\times n}$ to be a GCRD using the Smith normal form of the $m \times n$ compound matrix $P(\lambda)$ obtained by concatenating $P_i(\lambda)$ vertically, where $m=\sum_{i=1}^k m_i$. We also describe the complete set of degrees of freedom for the solution $G(\la)$, and we link it to the Smith form and Hermite form of $P(\la)$. 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 $P(\la)$, 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 $P(\la)$.