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Matrices carrées sur l’algèbre de Weyl. (Square matrices over the Weyl algebra). (French) Zbl 0655.16010
The Weyl algebra $$W_ n$$ is the algebra of all linear differential operators on $$C^ n$$ with polynomial coefficients. Two square matrices $$a\in M_{\ell}(W_ n)$$, $$a'\in M_{\ell '}(W_ n)$$ are said to be equivalent if for some $$r\geq 1$$ there exist $$b,c\in GL_ r(W_ n)$$ such that $$b(a\oplus I_{r-\ell})=(a'\oplus I_{r-\ell '})c$$ (here $$I_ p$$ is the identity matrix of order p). For example, any invertible matrix is equivalent to $$1\in W_ n$$. A matrix $$a\in M_{\ell}(W_ n)$$ is said to be deterministic if it is not a zero divisor. The main result of the paper claims that the equivalence class of a deterministic matrix always contains a $$2\times 2$$ matrix, but not, in general, a triangular matrix (of any order).
##### MSC:
 16S50 Endomorphism rings; matrix rings 16W20 Automorphisms and endomorphisms 32C38 Sheaves of differential operators and their modules, $$D$$-modules 58J15 Relations of PDEs on manifolds with hyperfunctions