Interlacing inequalities and control theory (Q1121211)

As it is well known the controllability indices and the invariant factors of a matrix pair (A,B) form a complete system of invariants for the state-feedback equivalence or block similarity. In this paper the following problem is solved: given an \(n\times n\) matrix A over an arbitrary field, under what conditions does there exist an \(n\times m\) matrix B such that the pair (A,B) is in a prescribed state-feedback equivalence class? The solution of this problem includes a characterization of the possible controllability indices of all controllable pairs (A,B) as B runs over the set of \(m\times n\) matrices. The solution of the first problem together with a result in an earlier paper of the author [ibid. 87, 113-146 (1987; Zbl 0632.15003)] is used to provide a new proof of the so-called Sá-Thompson interlacing inequalities for invariant factors [\textit{E. Marques de Sá}, ibid. 24, 33-50 (1979; Zbl 0395.15009); \textit{R. C. Thompson}, ibid. 24, 1-31 (1979; Zbl 0395.15003)], which characterize the matrices belonging to a prescribed similarity class and having a prescribed principal submatrix.