I was discussing applying Principal Component Analysis to a covariance matrix versus applying PCA to the corresponding correlation matrix with a collegue. This led me to think about the following question.

Given a covariance matrix $\Sigma \in \mathbb{R}^{n\times n}$ of random variables $X_1, \ldots, X_n$. From this covariance matrix we can calculate the correlation matrix $R \in \mathbb{R}^{n\times n}$ through $$R_{ij} = \frac{\Sigma_{ij}}{\sigma_{X_i}\sigma_{X_j}}$$ where $\sigma_{X_i}$ is the standard deviation of $X_i$.

What can we say about the eigenspace of $\Sigma$ in terms of $R$? Is there any way to relate the eigenvectors of $\Sigma$ to those of $R$?

If this is a trivial question then I apologize, a Google search hasn't given me any answers.