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

Abstract

A real square matrix $$Q$$ Q is a bilinear complementarity relation on a proper cone $$K$$ K in $$\mathbb{R }^n$$ R n if $$\begin{aligned} x\in K, s\in K^*,\,\,\text{ and }\,\,\langle x,s\rangle =0\Rightarrow x^{T}Qs=0, \end{aligned}$$ x ? K , s ? K ? , and ? x , s ? = 0 ? x T Q s = 0 , where $$K^*$$ K ? is the dual of $$K$$ K . The bilinearity rank of $$K$$ K is the dimension of the linear space of all bilinear complementarity relations on $$K$$ K . In this article, we continue the study initiated by Rudolf et al. (Math Prog Ser B 129:5---31, 2011). 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 $$K$$ K is the dimension of the Lie algebra of the automorphism group of $$K$$ 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.