Let $\Sigma$ be a hermitian positive definite matrix and $L$ be it's Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ is a diagonal matrix containing the real, positive eigenvalues of $\Sigma$, let us denote as $\sqrt{\Lambda}$ the diagonal matrix whose diagonal elements are the square roots of these eigenvalues.

What are some non trivial relationships between $L$ and $(P,\Lambda)$?

The one I have is: Since $(P\sqrt{\Lambda}P^\ast)$ is the unique positive square root of $\Sigma$ then $U = L^\ast P \Lambda^{-1/2} P^\ast$ is unitary.

... and that's about it. Maybe it counts as trivial?

Are there other interesting relations, maybe relations that take into account $L$'s triangular structure? In particular, I'd be interested in algorithm that derive $L$ from $(P,\Lambda)$ or vice-versa (and obviously which aren't merely the trivial composition of two algorithms).

Let $\Sigma$ be a hermitian positive definite matrix and $L$ be its Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ is a diagonal matrix containing the real, positive eigenvalues of $\Sigma$, let us denote as $\sqrt{\Lambda}$ the diagonal matrix whose diagonal elements are the square roots of these eigenvalues.

What are some non trivial relationships between $L$ and $(P,\Lambda)$?

The one I have is: Since $(P\sqrt{\Lambda}P^\ast)$ is the unique positive square root of $\Sigma$ then $U = L^\ast P \Lambda^{-1/2} P^\ast$ is unitary.

... and that's about it. Maybe it counts as trivial?

Are there other interesting relations, maybe relations that take into account $L$'s triangular structure? In particular, I'd be interested in algorithm that derive $L$ from $(P,\Lambda)$ or vice-versa (and obviously which aren't merely the trivial composition of two algorithms).