The change of similarity invariants under row perturbations: Generic cases (Q929475)

The authors consider the problem of characterizing the possible invariant factors or Weyr characteristic of all the matrices that can be obtained from a specified one by perturbing some of its rows. Given complex square matrices split as \[ M= \left( \begin{matrix} A & B \\ C & D \end{matrix} \right) \in \mathbb C^{(p+q)\times (p+q)}, \] the authors study the change of the similarity invariants of \(M\) under small additive perturbations on the last \(q\) rows. If \(\varepsilon >0\) is a real number, the authors obtain necessary and sufficient conditions that some polynomials, or some integer partitions, have to satisfy in order to be invariant factors, or the Weyr characteristic, of a \((p+q) \times (p+q)\) matrix \[ M'=\left( \begin{matrix} A & B \\ C' & D' \end{matrix} \right), \] where \(\left \| [C \;D]-[C' \;D'] \right \| < \varepsilon\). The authors give the solution of this problem in two significant generic particular cases, that is, when the pair \((A,B)\) is completely controllable and when \((A,B)\) is not completely controllable and the \(M\)-invariant subspace \[ {\mathcal S}={\mathcal C}(A,B) \times F^q=\left \{ [x \;y]^T \in F^{p+q} \;: \;x \in {\mathcal C}(A,B) \right \}, \] where \({\mathcal C}(A,B)\) is the controllability subspace of \((A,B)\) and \(F\) denotes any arbitrary field, is cyclic.