Linear transformations which leave controllable multiinput descriptor systems controllable (Q1824585)

The controllability of descriptor systems of the form Eẋ \(=\) Ax \(+\) Bu is considered. In particular, it is shown that the only linear functions \(f:\quad {\mathbb{C}}^{n,n+m}\to {\mathbb{C}}^{n,n+m}\) of the form \(f(X)=UXV\) \((U\in {\mathbb{C}}^{n,n}\), \(V\in {\mathbb{C}}^{n+m,n+m})\) which are controllability invariants satisfy: (i) U nonsingular, (ii) \(V=\left[ \begin{matrix} Q&0\\ 0&I \end{matrix} \right]\bar V\) where \(\bar V\) is a product of matrices of the types \(\left[ \begin{matrix} I&0\\ F&I \end{matrix} \right]\), \(\left[ \begin{matrix} I&0\\ 0&W \end{matrix} \right]\) and Q, W are nonsingular.