Consider real matrices. Is there an intrinsic measure on (normalized)symmetric semi positive definite(SPD) matrices? By intrinsic, I mean some measure derived from Haar measure of a group, since a SPD matrix can be generate from $M \in GL(n)$ as $MM^T$ or from anti-symmetic matrices by exponential map. Another possible connection to $O(n)$ is through Schur decomposition or so. Since SPD is not compact, we can normalize to "correlation coefficient" matrices. Let $D^2$ be the diagonal part of SPD $A$. We normalize $A$ as $D^{-1}AD^{-1}$, and it will have all 1's on diagonal.
Thanks!
Best Answer
Your space is the symmetric space for $GL(n, \mathbb{R}),$ with all that entails, or, if you prefer, the cone over the symmetric space for $SL(n, \mathbb{R}).$ See, for example, http://math.berkeley.edu/~reb/courses/261/18.pdf (but there is a million other references, including the venerable Helgason).