I am given $A$ a symmetric positive definite matrix, and $U$ which the Cholesky factor of $A$. I am also told that if $V$ is an upper triangular matrix such that $A$ = $V^TV$. I have to show that there exists a diagonal matrix $D$ whose entries on the main diagonal are either $−1$ or $1$ such that $V = DU$.
My thoughts are as follows. Based on the first statement, $A$=$U^T U$ as we are given that $U$ is a cholesky factor of $A$. Also based on this I can equate $A = V^T V = U^T U$, but I got stuck after this. How should I introduce $D$ in this and show that diagonal of $D$ should be $1$ or $-1$ to get $V = DU$?
Any help would be appreciated.