A real square matrix Q is a bilinear complementarity relation on a proper cone K in R^n if x in K, s in K^* with x perpendicular to s implies x^{T}Qs=0, where K^* is the dual of K. 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 by Rudolf et al. in Math. Programming, Series B, 129 (2011) 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 Schur-type results for Lyapunov-like transformations.

## Citation

trGOW11-05, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA, December 2011

## Article

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