On the stabilizer subgroup of a fair of matrices

AbstractLet (F,G) be a pair of matrices defined over an arbitrary field, Fn × n, Gn × m. Consider the natural action of GLn x GLm on this pair given by (F,G) ↦ (gFg-1,gGh-1), where (g,h) ∈ GLn × GLm. This action is of interest in system theory as well as the representation theory of quivers. In this paper we study the stabilizer subgroup of this action stab(F,G), i.e. {(g,h) ∈ GLn x GLm|gFg-1 = F,gGh-1 = G}