Let $A=A^T$ be a $n\times n$ positive definite matrix. Define the numbers $f_i(A)=\min\{j:a_{ij}\neq 0\}.$ Consider the following Cholesky decomposition: $A=LL^T$, with $L$ being a lower triangular matrix. I want to prove now that the following equality holds: $f_i(L)=f_i(A)$ for all $i=1,..,n$.
I am not sure how to start. So I was wondering if anyone could give me a tip or a start. This is how I tried to start, but the next steps led me nowhere:
$A=LL^T$ so we have $A^TL^{T^{-1}}=L$. Then I need to show that $f_i(LL^T)=f_i(A^TL^{T^{-1}})$ holds. I did not check if L and L^T are invertible but I can assume it.