Rational matrix pseudodifferential operators

The skewfield {Mathematical expression} of rational pseudodifferential operators over a differential field {Mathematical expression} is the skewfield of fractions of the algebra of differential operators {Mathematical expression}. In our previous paper, we showed that any {Mathematical expression} has a minimal fractional decomposition {Mathematical expression}, where {Mathematical expression}, and any common right divisor of {Mathematical expression} and {Mathematical expression} is a non-zero element of {Mathematical expression}. Moreover, any right fractional decomposition of {Mathematical expression} is obtained by multiplying {Mathematical expression} and {Mathematical expression} on the right by the same non-zero element of {Mathematical expression}. In the present paper, we study the ring {Mathematical expression} of {Mathematical expression} matrices over the skewfield {Mathematical expression}. We show that similarly, any {Mathematical expression} has a minimal fractional decomposition {Mathematical expression}, where {Mathematical expression} is non-degenerate, and any common right divisor of {Mathematical expression} and {Mathematical expression} is an invertible element of the ring {Mathematical expression}. Moreover, any right fractional decomposition of {Mathematical expression} is obtained by multiplying {Mathematical expression} and {Mathematical expression} on the right by the same non-degenerate element of {Mathematical expression}. 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. © 2013 Springer Basel.