Relationship between different equivalence relations in the space of standardizable systems

In the space of generalized time-invariant linear systems we consider two equivalence relations, generalizing block-similarity of pairs . Both equivalence relations can be defined by the action of Lie groups and acting on , Restricting ourselves to , the open and dense set of standardizable systems, we prove that α2 can be seen as the action of α1 on the orbit space . We also relate the corresponding miniversal deformations through the versal deformation of the orbit space.