Splitting multivariate normal into individual (correlated) components

Question

I have a multivariate normal variable $X$ with mean zero and variance $\Sigma$. I would like to write every component $X_i$ of $X$ as: $$ X_i = \phi_i Z_i $$ where $\phi_i$ is a scalar and $Z_i$ is a scalar standard normal random variable (for every $i$). Question: what are the values of $\phi_i$ (given $\Sigma$), and what is the correlation matrix of the random variables $Z$?

Answer 1

Using the standard properties of variance and covariance, we have that $\phi_i = \sqrt{\Sigma_{i,i}}$ and $$\displaystyle \operatorname{cov}(Z_i,Z_j)=\operatorname{cov}(\phi_i^{-1}X_i,\phi_j^{-1}X_j)= \frac{\Sigma_{i,j}}{\phi_i\phi_j}=\frac{\Sigma_{i,j}}{\sqrt{\Sigma_{i,i}}\sqrt{\Sigma_{j,j}}} = \rho_{i,j}$$ where $\rho_{i,j}$ is the correlation coefficient of $X_i$ and $X_j$ (and also of $Z_i$ and $Z_j$).