On the Uniform Nonsingularity Property for Linear Transformations on Euclidean Jordan Algebras

In a recent paper, Chua and Yi introduced the so-called uniform nonsingularity property for a nonlinear transformation on a Euclidean Jordan algebra and showed that it implies the global uniqueness property in the context of symmetric cone complementarity problems. In a related paper, Chua, Lin, and Yi raise the question of converse. In this paper, we show that, for linear transformations, the uniform nonsingularity property is inherited by principal subtransformations and, on simple algebras, it is invariant under the action of cone automorphisms. Based on these results, we answer the question of Chua, Lin, and Yi in the negative.