Rational matrix pseudodifferential operators

The skewfield $mathcal{K }(partial )$ of rational pseudodifferential operators over a differential field $mathcal{K }$ is the skewfield of fractions of the algebra of differential operators $mathcal{K }[partial ]$ . In our previous paper, we showed that any $Hin mathcal{K }(partial )$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,Bin mathcal{K }[partial ],,Bne 0$ , and any common right divisor of $A$ and $B$ is a non-zero element of $mathcal{K }$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-zero element of $mathcal{K }[partial ]$ . In the present paper, we study the ring $M_n(mathcal{K }(partial ))$ of $ntimes n$ matrices over the skewfield $mathcal{K }(partial )$ . We show that similarly, any $Hin M_n(mathcal{K }(partial ))$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,Bin M_n(mathcal{K }[partial ]),,B$ is non-degenerate, and any common right divisor of $A$ and $B$ is an invertible element of the ring $M_n(mathcal{K }[partial ])$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-degenerate element of $M_n(mathcal{K } [partial ])$ . 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.