Bounds on entries of Cholesky factors?

Question

Motivated by the problem of covariance estimation.

Let $\Sigma$ be a positive-definite matrix whose diagonal entries are identically 1. (i.e. $\Sigma$ is a correlation matrix.)

If $L$ is an lower-triangular matrix such that $L^T L = \Sigma$, can the entries of $L$ be bounded? (i.e. do there exist real numbers $l, u$ such that $l \geq (L)_{ij} \leq u$?)

Answer 1

OK, I'll post it as an answer. Since, by definition of $L$ and $\Sigma$, $\sum_i L_{i,j}^2 = \Sigma_{j,j} = 1,\,\,\forall j,$ we must have $|L_{i,j}| \le 1$.