A construction of non-isomorphic polyhedral cones using Lyapunov rank

Given >= 3 and 1 <=<-1, Gowda et al. in [On the bilinearity rank of a proper cone and Lyapunov-like transformations. Math Program. 2014;147(1-2, Ser. A):155-170], constructed a cone subset of R with at most n + 1 extreme vectors such that the Lyapunov rank of K, denoted by () is m. In this paper, for >= 3, when natural numbers (>), and m such that 1 <=<-1 are given, we construct a proper polyhedral cone with l extreme vectors and ()=. This construction results in non-isomorphic proper polyhedral cones with same number of extreme vectors (generators). Further, Lyapunov-like transformations on this cone are diagonal matrices of a specific type. We also prove that there is exactly one proper polyhedral cone (up to isomorphism) with four extreme vectors in R-3.