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 $$\mathbb{R }^n$$ if where $$K^*$$ is the dual of $$K$$ . The bilinearity rank of $$K$$ is the dimension of the linear space of all bilinear complementarity relations on $$K$$ . In this article, we continue the study initiated by Rudolf et al. (Math Prog Ser B 129:5-31, ). 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 Rudolf et al., compute the bilinearity ranks of symmetric and completely positive cones, and state some Schur-type results for Lyapunov-like transformations. [ABSTRACT FROM AUTHOR]