  The randomised Horn problem, in both its additive and multiplicative version,
has recently drawn increasing interest. Especially, closed analytical results
have been found for the rank-1 perturbation of sums of Hermitian matrices and
products of unitary matrices. We will generalise these results to rank-1
perturbations for products of positive definite Hermitian matrices and prove
the other results in a new unified way. Our ideas work along harmonic analysis
for matrix groups via spherical transforms that have been successfully applied
in products of random matrices in the past years. In order to achieve the
unified derivation of all three cases, we define the spherical transform on the
unitary group and prove its invertibility.
