Some P-properties for linear transformations on Euclidean Jordan algebras

A real square matrix is said to be a P-matrix if all its principal minors are positive. It is well known that this property is equivalent to: the nonsign-reversal property based on the componentwise product of vectors, the order P-property based on the minimum and maximum of vectors, uniqueness property in the standard linear complementarity problem, (Lipschitzian) homeomorphism property of the normal map corresponding to the nonnegative orthant. In this article, we extend these notions to a linear transformation defined on a Euclidean Jordan algebra. We study some interconnections between these extended concepts and specialize them to the space S-n of all n x n real symmetric matrices with the semidefinite cone S-+(n) and to the space R-n with the Lorentz cone. (C) 2004 Elsevier Inc. All rights reserved.