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.

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