I've been wondering about the following:
Conjecture: If $A$ is a (strictly) diagonally dominant symmetric positive definite matrix, and $A=LDL^T$ is its square-root free Cholesky factorization, the $L_{ij}\leq 1$ for all $i,j$.
This property clearly does not hold in general, in particular if $A$ is not diagonally dominant. How could I go about proving the Conjecture (or is this already known)?
I would be satisfied if the Conjecture were true for a matrix $A$ with just $0$ or $1$ everywhere off the diagonal, and some (large) $M$ in every diagonal entry.
