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

### M. Seetharama Gowda and
J. Tao

### Technical Report Number: trGOW11-05

### Abstract:

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