On minimal positive basis for polyhedral cones in n?>

Abstract

For polyhedral cones in the Euclidean space, we present its conic dimension, which is invariant under linear isomorphisms that is sensitive to the number of generators of this cone, and the related notion of conic basis. We may interpret these two notions as versions of the definitions of linear dimension and linear basis for linear subspaces in the setting of polyhedral cones. We establish a conic version of the rank-nullity theorem that, in this case, is an inequality involving the conic dimensions of both the cone and its image under a linear map. We use this conic rank-nullity inequality to establish both a decomposition and a union of conic basis, involving the lineality space of the cone. We introduce the signature of a polyhedral cone and establish some results on the injectivity of a linear map and the preservation of the signature of a polyhedral cone under linear maps. In particular, we show that a linear map that acts injectively on the linear span of a polyhedral cone preserves its signature.