A Matsumoto–Yor property for Kummer and Wishart random matrices

For a positive integer rr, let II denote the r×rr×r unit matrix. Let XX and YY be two independent r×rr×r real symmetric and positive definite random matrices. Assume that XX follows a Kummer distribution while YY follows a non-degenerate Wishart distribution, with suitable parameters. This note points out the following observation: the random matrices U:=[I+(X+Y)−1]1/2[I+X−1]−1[I+(X+Y)−1]1/2U:=[I+(X+Y)−1]1/2[I+X−1]−1[I+(X+Y)−1]1/2 and V:=X+YV:=X+Y are independent and UU follows a matrix beta distribution while VV follows a Kummer distribution. This generalizes to the matrix case an independence property established in Koudou and Vallois (2010) for r=1r=1.