The distribution of symmetric matrix quotients

Phillips (J. Multivariate Anal. 16 (1985) 157) generalizes Cramerʹs (Mathematical Methods of Statistics, Princeton University Press, Princeton, NJ, 1946) inversion formula for the distribution of a quotient of two scalar random variables to the matrix quotient case. However, he gives the result for the asymmetric matrix quotient case. This note extends Phillips’ (1985) result to the symmetric matrix quotient case.