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 , , with coefficients in a generic field and with common column dimension . We give necessary and sufficient conditions for a matrix to be a GCRD using the Smith normal form of the compound matrix obtained by concatenating vertically, where . We also describe the complete set of degrees of freedom for the solution , and we link it to the Smith form and Hermite form of . We then give an algorithm for constructing a particular minimum size solution for this problem when or , using state-space techniques. This new method works directly on the coefficient matrices of , 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 .