Mostow-Type Decompositions and Geometric Means for Symmetric Cones

Motivated by Mostow's decomposition theorem for positive definite matrices and by subsequent matrix factorizations involving geometric means, we establish a Mostow-type decomposition for arbitrary symmetric cones. As a consequence, we derive an analogous decomposition for the automorphism group of the cone. We also introduce a natural Hadamard metric on the symmetric cone by pulling back an $\ell^2$-product metric, together with its midpoint operation and associated Karcher mean.