Jordan Algebras and applications to the mixed linear models

We present the properties of Jordan Algebras as a subspace of real sym- metric matrices definite in 1934 by Jordan and co-authors (see (8)) or quadratic subspaces defined independently by Seely in 1971 (see (9)) . The full char- acterization of such subspaces will be given. Moreover, some property for statistical inferences in mixed linear models will be presented. It will be shown that optimal estimators for parameters in mixed linear models and existence of optimal tests can be characterized in terms of Jordan Algebras. The presentation is given using coordinate free approach. Some selected pa- pers using this approach are given below.