Suppose that the random vector (X 1 , ..., X q ) follows a Dirichlet distribution on R q + with parameter ( p 1 , ..., p q ) # R q + . For f 1 , ..., f q >0, it is well-known that
. In this paper, we generalize this expectation formula to the singular and non-singular multivariate Dirichlet distributions as follows. Let 0 r denote the cone of all r_r positive-definite real symmetric matrices. For x # 0 r and 1 j r, let det j x denote the j th principal minor of x. For s=(s 1 , ..., s r ) # R r , the generalized power function of x # 0 r is the function 2 s (x)=(det 1 x) s 1 &s 2 (det 2 x) s 2 &s 3 } } } (det r&1 x) s r&1 &s r (det r x) s r ; further, for any t # R, we denote by s+t the vector (s 1 +t, ..., s r +t). Suppose X 1 , ..., X q # 0 r are random matrices such that (X 1 , ..., X q ) follows a multivariate Dirichlet distribution with parameters p 1 , ..., p q . Then we evaluate the expectation E [2 s 1 (X 1 ) } } } 2 s q (X q ) 2 s 1 + } } } +s q + p ((a+ f 1 X 1 + } } } + f q X q ) &1 )], where a # 0 r , p= p 1 + } } } + p q , f 1 , ..., f q >0, and s 1 , ..., s q each belong to an appropriate subset of R r + . The result obtained is parallel to that given above for the univariate case, and remains valid even if some of the X j 's are singular. Our derivation utilizes the framework of symmetric cones, so that our results are valid for multivariate Dirichlet distributions on all symmetric cones.