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).
Best Answer
First, let me rephrase your remark. Let $L=HU$ be the polar factorization of $L$ ($H$ hermitian positive definite, $U$ unitary). Then $\Sigma=LL^\ast=H^2$ tells you that the Hermitian part of $L$ is $\sqrt\Sigma$. Then $U=L\Sigma^{-1/2}$ is its unitary part.
On the other hand, you have $\sqrt\Sigma=LQ^\ast=QL^\ast$. This is exactly the $QR$-factorization of $\sqrt\Sigma$: $L^\ast$ is the upper triangular part with positive real diagonal in the $QR$-factorization of $\sqrt\Sigma$.