A constructive version of Fitting´s theorem on isomorphisms and equivalences of linear systems

Within the algebraic analysis approach to linear system theory, a multidimensional linear system can be studied by means of its associated finitely presented left module. Testing whether two linear systems/modules are isomorphic (the so-called equivalence problem) is an important issue in system/module theory. In this paper, we explicitly characterize the conditions for a homomorphism between two finitely presented left modules to define an isomorphism, and we give an explicit formula for the inverse of an isomorphism. Then, we constructively study Fitting´s major theorem, which shows how to enlarge matrices presenting isomorphic modules by blocks of 0 and I to get equivalent matrices. The consequences of this result on the Auslander transposes and adjoints of the finitely presented left modules are given. The different results developed in this paper are implemented in the OREMORPHISMS package.