On constructing nearly decomposable matrices

Abstract

In this paper we consider the construction of nearly decomposable matrices from nearly decomposable matrices of smaller dimension. In particular, if ${A_1},{A_2}, \cdots ,{A_s}$ are nearly decomposable matrices we consider the problem of finding matrices ${E_1},{E_2}, \cdots ,{E_s}$ each with exactly one nonzero entry so that \[ \left [ {\begin {array}{*{20}{c}}{{A_1}} & 0 & { \cdots 0} & {{E_1}} \\ {{E_2}} & {{A_2}} & { \cdots 0} & 0 \\ \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & { \cdots {E_s}} & {{A_s}} \\ \end {array} } \right ]\] is nearly decomposable.