The change of feedback invariants under one row perturbation (Q875011)

The authors characterize possible feedback invariants of a rectangular matrix under small additive perturbations on one of its rows. Different authors have described the possible similarity invariants of a square matrix under small additive perturbations on one of its rows. Also, they have described the possible feedback invariants of matrices obtained by addition of a complex matrix with sufficiently small entries to an arbitrary rectangular complex matrix. This paper is a natural prolongation of those results. Let \(A \in \mathbb C^{n \times n}\), \(\mathbb C \in \mathbb C^{n \times m}\), \(b^T \in \mathbb C^{n \times 1}\), \(a \in \mathbb C^{1 \times n}\), \(c \in \mathbb C^{1 \times m}\) and \(x \in \mathbb C^{1 \times 1}\); and consider the following rectangular matrix \[ M= \left( \begin{matrix} A & b^T & C \\ a & x & c \end{matrix} \right). \] In this paper the authors study the feedback invariants of \(M\) under small perturbations on its row \([a, x, c]\). They obtain necessary and sufficient conditions in order to assure that in every neighbourhood of \(M\) there exists a matrix \[ M^{\prime}= \left( \begin{matrix} A & b^T & C \\ a' & x' & c' \end{matrix} \right) \] with a feedback invariants prescribed.