Underlying algebraic framework of equivalence relations on linear systems

The algebraic framework that underlies the various concepts of equivalence for linear systems is examined via the introduction of an equivalence relation that has as special cases system equivalence, strict system equivalence and local system equivalence. Another equivalence relation, unstable system equivalence, on the set of polynomial realizations of a rational matrix is also introduced and studied as an illustrative example. The concept of unstable system equivalence should prove useful, since it ensures that the behaviour of the dynamical systems, associated with the polynomial realizations, is preserved at all the finite unstable frequencies, but does not attempt to preserve the behaviour of the dynamical systems at the finite stable frequencies. The relationship between local system equivalence and system equivalence at infinity is also described.