Cholesky matrix properties

Question

Given the Cholesky decomposition $U U^T = A$ of a positive definite matrix, is there anything known about $U$ itself? It is obviously not symmetric and also not positive definite but does it have any special properties about the trace, determinant, eigenvalues, diagonal dominant, etc?

Answer 1

The Choleksy Factorization from Trefethan is given where $$ A \in \mathbb{C}^{m \times m} $$ is Hermitian positive definite then we have $$ A = R^{*}R, r_{jj} > 0 $$ where R is an upper triangular matrix.

Does it have any special properties?

It is pretty well known that determinant of triangular matrices is the product of the diagonal that is the following

$$ det(R) = \prod_{i=1}^{m} r_{ii} $$

Also one should note that eigenvalues of the triangular matrix are entries on the main diagonal. Then we have the following. $$ det(R) = \prod_{i=1}^{m} \lambda_{i}$$

Also the following

$$ tr(R) = \sum_{i=1}^{m} \lambda_{i} $$

Answer 2

The definition I know of Cholesky decomposition (see here for instance) includes that $U$ is a lower triangular matrix with real and positive diagonal entries. So from here you can deduce information about determinant and eigenvalues.