On Transformations and Determinants of Wishart Variables on Symmetric Cones

Let x and y be independent Wishart random variables on a simple Jordan algebra V. If c is a given idempotent of V, write $$x = x_1 + x_{12} + x_0 $$ for the decomposition of x in $$V(c,1) \oplus V(c,1/2) \oplus V(c,0)$$ where V(c,λ) equals the set of v such that cv=λv. In this paper we compute E(det(ax+by)) and some generalizations of it (Theorems 5 and 6). We give the joint distribution of (x 1, x 12, y 0) where $$y_0 = x_0 - P(x_{12} )x_1^{ - 1} $$ and P is the quadratic representation in V. In statistics, if x is a real positive definite matrix divided into the blocks x 11, x 12, x 21, x 22, then y 0 is equal to $$x_{22.1} = x_{22} - x_{21} x_{11}^{ - 1} x_{12} $$ . We also compute the joint distribution of the eigenvalues of x (Theorem 9). These results have been known only when V is the algebra of Hermitian matrices with entries in the real or the complex field. To obtain our results, we need to prove several new results on determinants in Jordan algebras. They include in particular extensions of some classical parts of linear algebra like Leibnitz's determinant formula (Proposition 2) or Schur's complement (Eqs. (3.3) and (3.6)).