Consider the space of $n \times n$ positive definite symmetric matrices and let $\Sigma$ be one such matrix. We make this space into a Riemannian manifold $M$ by means of the metric $$ds^2=tr(\Sigma^{-1}d\Sigma\Sigma^{-1}d\Sigma)$$ Now fix a $p<n$ and consider the collection $\Phi_p$ of all positive definite symmetric matrices which can be written $$B^TB+D$$ for some $B$ and $D$ where $B$ is $p \times n$ and $D$ is a diagonal positive definite matrix. Here $B$ and $D$ are allowed to vary but $p$ must stay fixed. Is $\Phi_p$ an embedded submanifold of $M$ and if so how can I put the induced metric on it?

## 1 Answer

This Riemannian metric on the full space of all positive definite matrices turns up in the paper (and others)

There are explicit formulas for geodesics and for curvature. There is only one incomplete geodesic, but the space if far from being geodesically convex.

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