# Variance-covariance matrix as the sum of variance covariance matrices

I have a variance-covariance matrix, $\mathrm{V}$. This allows me to take a vector, $x$ of independent random variables drawn from a known distribution, and induce a required variance-covariance structure to give a correlated random variable $y$ from:

$y = \mathrm{V}^{\frac{1}{2}} x$

Because $\mathrm{V}$ is positive definite, I can find $\mathrm{V}^{\frac{1}{2}}$ using the Cholesky decomposition.

However, I know that the variance-covariance structure represented by $\mathrm{V}$ actually comprises three separate components - say $\mathrm{V}_1$, $\mathrm{V}_2$ and $\mathrm{V}_3$.

It would be highly useful to show the impact of these three separate components. Is there any way to do this?

• Please define "the impact of these three separate components." – Did Nov 5 '14 at 11:37