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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
.
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