Computing the nearest stable matrix pairs
Nicolas Gillis, Volker Mehrmann, Punit Sharma · Numerical Linear Algebra with Applications · 2018
Summary In this paper, we study the nearest stable matrix pair problem: given a square matrix pair ( E , A ), minimize the Frobenius norm of (Δ E ,Δ A ) such that ( E +Δ E , A +Δ A ) is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian matrix pairs: A matrix pair ( E , A ) is dissipative Hamiltonian if A =( J − R ) Q with skew‐symmetric J , positive semidefinite R , and an invertible Q such that Q T E is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.