On the existence of linear systems with prescribed invariants for system similarity

We define the local Wiener–Hopf, controllability and Hermite indices of nonsingular polynomial matrices and controllable matrix pairs and deduce that the local indices of matrix pairs are the local indices of their polynomial matrix representations. We solve the problem of the existence of nonsingular polynomial matrices with prescribed invariant factors and local and global, either Hermite or Wiener–Hopf, indices. Finally, we apply this result to finding necessary and sufficient conditions for the existence of a controllable pair (A, B) with prescribed invariant factors for A and local and global, either Hermite or controllability, indices.