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 $P_i(\lambda )\in \text{F}[\text{la}{]}^{m_i\times n}$, $i=1,\dots ,k$ with coefficients in a generic field $\text{F}$, and with common column dimension $n$. We give necessary and sufficient conditions for a matrix $G(\text{la})\in \text{F}[\text{la}{]}^{\mathrm{\ell }\times n}$ 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 $G(\text{la})$, and we link it to the Smith form and Hermite form of $P(\text{la})$. We then give an algorithm for constructing a particular minimum rank solution for this problem when $\text{F}=\text{C}$ or $\mathbb{R}$, using state-space techniques. This new method works directly on the coefficient matrices of $P(\text{la})$, 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 $P(\text{la})$.