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.