For $\sum$ covariance matrix ($n\times n$, symmetric and positive definite) and $C$ lower triangular matrix with real and positive entries, how can I prove the equivalence as from title?
I know that the inverse of a lower triangular matrix is again lower triangular. I also know that $(C^T)^{-1}=(C^{-1})^T$. Thus:
$\sum=CC^T \Rightarrow \sum^{-1}=(CC^T)^{-1}=(C^{-1})(C^T)^{-1}=(C^{-1})(C^{-1})^T=CC^T$ with abuse of notation.
But I have to obtain that $\sum^{-1}=C^TC$. Where I wrong?