  In this paper, we study the nearest stable matrix pair problem: given a
square matrix pair $(E,A)$, minimize the Frobenius norm of
$(\Delta_E,\Delta_A)$ such that $(E+\Delta_E,A+\Delta_A)$ is a stable matrix
pair. We propose a reformulation of the problem with a simpler feasible set by
introducing dissipative Hamiltonian (DH) matrix pairs: A matrix pair $(E,A)$ is
DH if $A=(J-R)Q$ with skew-symmetric $J$, positive semidefinite $R$, and an
invertible $Q$ such that $Q^TE$ 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.
