Matrix extensions of Liouville-Dirichlet-type integrals

Abstract

The Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={ x 1,…, x k : x i ⩾0, i=1,…, k, s⩽∑ k 1 x i ⩽ T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes Ω={V 1,…,V k: V i>0, i=1,…,k, 0⩽∑V i⩽t} , where now each V i is a p× p symmetric matrix and A⩾ B means that A− B is positive semidefinite.