On the bilinearity rank of a proper cone and Lyapunov-like transformations

A real square matrix Q is a bilinear complementarity relation on a proper cone K in R if x ∈ K, s ∈ K∗, and 〈x, s〉 = 0⇒ xQs = 0, where K∗ is the dual of K [14]. The bilinearity rank of K is the dimension of the space of all bilinear complementarity relations on K. In this article, we continue the study initiated in [14] by Rudol et al. We show that bilinear complementarity relations are related to Lyapunov-like transformations that appear in dynamical systems and in complementarity theory and further show that the bilinearity rank of K is the dimension of the Lie algebra of the automorphism group of K. In addition, we correct a result of [14], compute the bilinearity ranks of symmetric and completely positive cones, and state Schur-type results for Lyapunov-like transformations.