$P\in\mathbb{R}^{n \times n}$ is a symmetric positive definite (SPD) covariance matrix, and can be factorized into standard deviations and correlations as $P=\sigma \rho \sigma^T=\sigma \rho \sigma$. Here, $\sigma \in\mathbb{R}^{n \times n}$ is a diagonal matrix containing the standard deviations, $\sigma = diag(\sqrt{diag(P)})$, and the correlation matrix $\rho \in\mathbb{R}^{n \times n}$ has 1 on its diagonals and $\rho_{i,j}\in(-1,1]$.
Is it possible to relate the SVD of $\rho$ to the SVD of $P$? That is, given $\rho=U_\rho D_\rho U_\rho^T$ and $\sigma$, can I relate these to the SVD of $P$? The benefit is numerical stability since $\rho$ is bounded between -1 and 1.
Example: $P=\begin{bmatrix} 10^{6} & 0.05 \\ 0.05 & 10^{-8} \end{bmatrix}=\sigma \rho \sigma$ where $\rho=\begin{bmatrix} 1 & 0.5 \\ 0.5 & 1 \end{bmatrix}$ and $\sigma=\begin{bmatrix} 10^3 & 0 \\ 0 & 10^{-4} \end{bmatrix}$. Given the SVD of $\rho$ and $\sigma$, how do I compute the SVD of $P$?