SVD - reconstruction from U,S,V

Question

I am learning some linear algebra for image compression and I am stuck at this point:

Suppose I have a matrix $R$,

$$ \begin{bmatrix} 5 & 7\\ 2 & 1\\\end{bmatrix} $$

Then I compute the covariance matrix s.t. $$\Sigma =\frac12R^TR$$ And I performed SVD with a Matlab function s.t. $ [U, S, V] = svd(\Sigma) $

I can see that $USV = \Sigma$ but how can I solve this equation below for $R$: $\Sigma=\frac12R^TR$

Answer 1

Given $\Sigma = \frac{1}{2}R^TR$, you first let $\Sigma = LL^T$. Find out $L$ by Cholesky factorization.

Now $(\frac{1}{\sqrt{2}})^2(R^T)(R^T)^T=LL^T$. Can you continue from here?