The Unmixing Problem

High-gain adaptive stabilization of minimum-phase linear systems raises the so-called unmixing problem: finding a finite set A of matrices, with the property that for any M ɛ GL (n), σ( AM ) ⊂C ₊ for at least one A $A∈A$ . First, some preliminary results will be given, leading up to a non-constructive proof of the existence of such finite A's. For the cases n = 2 and 3, a fairly satisfying solution is presented. In higher dimensions, the situation is more complicated, and negative results are given. Based on the Euler angles for δO(n), an explicit solution is given for arbitrary n .