I need to find the correlation matrix from a covariance matrix. The covariance matrix is the following: $$\frac{1}{4} \begin{pmatrix} 1 & 0.1 \\ 0.1 & 2 \end{pmatrix}$$
I've read this post but I still can't get how for $a_{ij}$ in the correlation matrix $a_{ij} = \frac{\sigma_{ij}}{\sigma_i \sigma_j}$. How do we get a single $\sigma_i$ or $\sigma_j$? Thanks for the help.
Best Answer
$\sigma_{ii}$ is the covariance of the $i$-th component with itself, and if you inspect the definition of covariance, you'll see that it is actually equal to the variance of the $i$-th component. $\sigma_i$ is its standard deviation, hence $\sigma_i=\sqrt{\sigma_{ii}}$. So in your matrix, $\sigma_1 = \sqrt{\frac14\cdot 1} =\frac12$ and $\sigma_2 = \sqrt{\frac14\cdot 2} =\frac1{\sqrt2}$.