Rational matrix pseudodifferential operators

Abstract

The skewfield K(∂) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[∂]. In our previous paper, we showed that any H ∈ K(∂) has a minimal fractional decomposition H = AB[superscript −1] , where A,B ∈ K[∂], B ≠ 0, and any common right divisor of A and B is a non-zero element of K . Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-zero element of K[∂] . In the present paper, we study the ring M[subscript n](K(∂)) of n × n matrices over the skewfield K(∂). We show that similarly, any H ∈ M[subscript n](K(∂)) has a minimal fractional decomposition H = AB[superscript −1], where A,B ∈ M[subscript n](K[∂]), B is non-degenerate, and any common right divisor of A and B is an invertible element of the ring M[subscript n](K[∂]). Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-degenerate element of M[subscript n](K[∂]). We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.