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We generalize the following univariate characterization of the Kummer andGamma distributions to the cone of symmetric positive definite matrices: let$X$ and $Y$ be independent, non-degenerate random variables valued in $(0,\infty)$, then $U= Y/(1+X)$ and $V = X(1+U)$ are independent if and only if $X$follows the Kummer distribution and $Y$ follows the the Gamma distribution withappropriate parameters. We solve a related functional equation in the cone ofsymmetric positive definite matrices, which is our first main result and applyits solution to prove the characterization of Wishart and matrix-Kummerdistributions, which is our second main result.

Independence characterization for Wishart and Kummer random matrices