Relationship between different equivalence relations in the space of standardizable systems

In the space MM of generalized time-invariant linear systems Ex˙=Ax+Bu we consider two equivalence relations, generalizing block-similarity of pairs (A,B)∈Mn(C)×Mn×m(C)(A,B)∈Mn(C)×Mn×m(C). Both equivalence relations can be defined by the action of Lie groups G1=Gl(n;C)×Gl(m;C)×Mm×n(C)×Mm×n(C)G1=Gl(n;C)×Gl(m;C)×Mm×n(C)×Mm×n(C) and G2=Gl(n;C)×G1G2=Gl(n;C)×G1 acting on MM,α1:G1×M→M((P,R,U,V),(E,A,B))→(P-1EP+P-1BU,P-1AP+P-1BV,P-1BR)α2:G2×M→M((P,Q,R,U,V),(E,A,B))→(QEP+QBU,QAP+QBV,QBR).Restricting ourselves to SS, the open and dense set of standardizable systems, we prove that α2 can be seen as the action of α1 on the orbit space S/Gl(n;C)S/Gl(n;C). We also relate the corresponding miniversal deformations through the versal deformation of the orbit space.