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

M. Seetharama Gowda and J. Tao

Technical Report Number: trGOW11-05


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^*, and x orthogonal to s implies that 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 in a recent paper of Rudolf 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 Rudolf et al, compute the bilinearity ranks of symmetric and completely positive cones, and state Schur-type results for Lyapunov-like transformations.

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AMS 2012

chennai conf