2.12 Suppose V: p × p is positive definite and V is partitioned as where V11: q × q and V11 is positive definite. Define ƒ ≡ ∫ V (n–p–1)/2dV12, where n > p + 1, and the integration extends over all possible values of the elements of V12. Using (2.6.7), (2.15.2), and (2.15.3), show that ƒ = C V11 (n–q–1)/2 V22 (n–(p–q)–1)/2, where C is a numerical constant depending upon n, p, and q only. It will be seen in Corollary (5.1.3) that this result may be used to show that when V is a random matrix following a Wishart distribution (5.1.1) with block diagonal scale matrix, V11 and V22 are independently distributed. [Remark: A block diagonal matrix is one whose principal submatrices have possibly nonzero elements, but all other elements are zero.]