If the positive definite matrix P forms a manifold, is that the subset that {P: P = V V^T + D} where V is a low rank matrix and D is a positive definite matrix a sub-manifold?
This idea is corresponding to an application in finance that P is covariance matrix and V is a factor structure.
Thanks a lot.
What a coincidence. Someone asked a closely related (but different) question just two days ago.
Let $k\le n$ be a positive integer. Note that the five sets \begin{align*} \mathcal{P}&=\{P\in M_n(\mathbb{R}): P \text{ is positive definite}\},\\ \mathcal{S}_1&=\{D+VV^T: D\in\mathcal{P},\, V\in M_n(\mathbb{R}),\, \operatorname{rank} V\le k\},\\ \mathcal{S}_2&=\{D+VV^T: D\in\mathcal{P},\, V\in M_n(\mathbb{R}),\, \operatorname{rank} V=k\},\\ \mathcal{S}_3&=\{D+VV^T: D\in\mathcal{P},\, V\in M_{n,k}(\mathbb{R}),\, \operatorname{rank} V\le k\},\\ \mathcal{S}_4&=\{D+VV^T: D\in\mathcal{P},\, V\in M_{n,k}(\mathbb{R}),\, \operatorname{rank} V=k\},\\ \end{align*} are identical to each other. So, whatever properties does $\mathcal{P}$ possess, those $\mathcal{S}_j$s possess the same properties too.