Polynomial matrix primitive factorization over arbitrary coefficient field and related results

Morf, Levy, and Kung and Youla and Gnavi presented a primitive factorization algorithm which extracts in some sense the content of a (full rank) matrix
A
with entries in the ring
K[z,\omega]
of bivariate polynomials over some field
K
. However, the algorithms presented in both cases specify and require the coefficient field
K
to be algebraically closed-typically the field of complex numbers. It is desirable, from theoretical and computational standpoints, to have no such restriction on
K
; so, for example, one could do the factorization over the real field or even the field of rational numbers, provided the coefficients start out in these fields. Here an algorithm which produces a primitive factorization over an arbitrary field
K
is presented and the use of this algorithm is illustrated by a nontrivial example. Several related results leading to a general factorization theorem are stated and proved. Scopes for applying the results in various problems of scientific and engineering interest are mentioned.