Relationship between the eigenvalues of a matrix and its Cholesky decomposition

Question

Cholesky decomposition states that if $A$ is symmetric positive semidefinite matrix , then there exists a lower triangular matrix $L$ with nonnegtive diagonal elements such that

$$ A = LL^T $$

Is there any relation between the eigenvalues of $A$ and the elements (not the eigenvalues) of $L$?

Answer 1

I don't think there is any particular relationship between them. You could, however, state some trivial facts, for example:

1. $A \geq 0 \Rightarrow l_{ii}\geq0$
2. $\prod \lambda_i = \prod l_{ii}^2 $

Also, note that you cannot take $L=UD^{1/2}$ ($UDU^T$ being the eigendecomposition of $A$), because there is no guarantee for $L$ to be lower-triangular (which must be the case for Cholesky factorization).

Cholesky decomposition is only a concise representation of Gaussian elimination and my understanding is that, it is too weak to reveal any information concerning the eigenstructure of $A$.