On estimating the separation between (A,B) and (C,D) associated with the generalized Sylvester equation AXD−BXC=E

In this paper, we estimate the following quantity sep[(A,B),(C,D)]=min‖X‖F=1‖AXD−BXC‖F,where A,B∈Cm×m,C,D∈Cn×n and ‖⋅‖F is the Frobenius norm. In terms of the generalized Schur decomposition and Weiestrass canonical form of the regular matrix pair, lower and upper bounds of sep[(A,B),(C,D)] are derived, respectively. The results are illustrated by numerical examples.