  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 $P(\la)$.
