Some new results on the P -type properties of Z -transformations on symmetric cones

This article is on Z-transformations with respect to a symmetric cone in a Euclidean Jordan algebra. Motivated by a well known result on Z matrices (i.e. matrices with off diagonal entries are non-positive), we show that various P-type properties are equivalent for a Z-transformation. These P-type properties arise from the theory of linear complementarity problems. Precisely, by utilizing the concept of principal subtransformations in a Euclidean Jordan algebra, we show that for a Z-transformation, the so-called completely Q, completely P and positive principal minor properties are equivalent. Examples of Z-transformations with completely P-property are then given. These examples are constructed by obtaining some new results in linear complementarity problems which are of independent interest.