  If $\alpha,\beta>0$ are distinct and if $A$ and $B$ are independent
non-degenerate positive random variables such that
  $$S=\tfrac{1}{B}\,\tfrac{\beta A+B}{\alpha A+B}\quad \mbox{and}\quad
T=\tfrac{1}{A}\,\tfrac{\beta A+B}{\alpha A+B}
  $$ are independent, we prove that this happens if and only if the $A$ and $B$
have generalized inverse Gaussian distributions with suitable parameters.
Essentially, this has already been proved in Bao and Noack (2021) with
supplementary hypothesis on existence of smooth densities.
  The sources of these questions are an observation about independence
properties of the exponential Brownian motion due to Matsumoto and Yor (2001)
and a recent work of Croydon and Sasada (2000) on random recursion models
rooted in the discrete Korteweg - de Vries equation, where the above result was
conjectured.
  We also extend the direct result to random matrices proving that a matrix
variate analogue of the above independence property is satisfied by independent
matrix-variate GIG variables. The question of characterization of GIG random
matrices through this independence property remains open.
