On the class of reduced order models obtainable by projection

The paper investigates the properties of general reduced order models obtained by projection of a higher order system. It answers questions such as, are any two models of different orders related by a projection? Is it possible to obtain the same reduced order model using different projections? How to find, if it exists, a projection that relates the two models?, etc. It is shown that answers to those questions can be obtained by investigating the properties of a certain matrix pencil. The key tool is the Kronecker canonical form, and in some cases of square systems the problem becomes that of generalized eigenvalues.