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This paper is a sequel to "Computing diagonal form and Jacobson normal formof a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. Wepresent a new fraction-free algorithm for the computation of a diagonal form ofa matrix over a certain non-commutative Euclidean domain over a computablefield with the help of Gr\"obner bases. This algorithm is formulated in ageneral constructive framework of non-commutative Ore localizations of$G$-algebras (OLGAs). We split the computation of a normal form of a matrixinto the diagonalization and the normalization processes. Both of them can bemade fraction-free. For a matrix $M$ over an OLGA we provide a diagonalizationalgorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$holds and $D$ is diagonal. The fraction-free approach gives us more informationon the system of linear functional equations and its solutions, than theclassical setup of an operator algebra with rational functions coefficients. Inparticular, one can handle distributional solutions together with, say,meromorphic ones. We investigate Ore localizations of common operator algebrasover $K[x]$ and use them in the unimodularity analysis of transformationmatrices $U,V$. In turn, this allows to lift the isomorphism of modules over anOLGA Euclidean domain to a polynomial subring of it. We discuss the relation ofthis lifting with the solutions of the original system of equations. Moreover,we prove some new results concerning normal forms of matrices over non-simpledomains. Our implementation in the computer algebra system {\scSingular:Plural} follows the fraction-free strategy and shows impressiveperformance, compared with methods which directly use fractions. Since weexperience moderate swell of coefficients and obtain simple transformationmatrices, the method we propose is well suited for solving nontrivial practicalproblems.

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases