On the stability of a convex set of matrices

In this paper we give an alternative proof of the constant inertia theorem for convex compact sets of complex matrices. It is shown that the companion matrix whose non-trivial column is negative satisfies the directional Lyapunov condition (inclusion) for real multiplier vectors. An example of a real matrix polytope that satisfies the directional Lyapunov condition for real multiplier vectors and which has nonconstant inertia is given. A new stability criterion for convex compact sets of real Z-matrices is given. This criterion uses only real vectors and positive definite diagonal matrices.