# A submanifold of the space positive definite matrices

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?

• Why is the metric a metric? – Igor Rivin Nov 22 '13 at 19:52
• In fact, if $X_1 = X_2 = I,$ the distance is not zero, which is a little sad. – Igor Rivin Nov 22 '13 at 19:58
• So the first thing to test is: say $P$, $Q$ come from this diag+low-rank set. Now, is the geometric mean of $P$ and $Q$ also in this set? If not, then this won't be a submanifold (in the sense that you are searching for)... – Suvrit Nov 23 '13 at 16:36
• It's not a smooth submanifold in general. Just look at the case $(n,p)=(3,1)$, where $\Phi_1$ contains an open set, but is not open: For $a\in\Phi_1$, you have to have $S := a_{12}a_{23}a_{31}\ge0$. If $S>0$, then you only need that $a_{ij}^2a_{kk} > S$ for all $(i,j,k)$ a permutation of $(1,2,3)$ in order for $a$ to lie in $\Phi_1$. Thus, $\Phi_1$ contains an open set. However, $\Phi_1$ clearly doesn't contain a neighborhood of the identity matrix (which does lie in $\Phi$), since $S$ is not positive at the identity, so it's not an open set. – Robert Bryant Nov 24 '13 at 11:16
• @IgorRivin: Thanks. Actually, I think that, properly worded, there is a somewhat interesting question here. The OP really should be asking about the structure of this subvariety of 'low-rank' perturbations of a flat submanifold (i.e., the diagonal matrices) of the space of quadratic forms. It's a natural subvariety of this symmetric space, and it's conceivable that knowing something about this space has applications in information geometry. – Robert Bryant Nov 25 '13 at 6:01