# LU decomposition for orthogonal or unitary matrices?

Is there any references on LU decomposition for orthogonal or unitary matrices?

It seems to me that the diagonal entries of $$U$$ has some nice structure regarding to the Euler angles of the original matrix. As one can easily see under a Euler parametrisation:

$$\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}=\begin{bmatrix}1&0\\\tan\theta&1\end{bmatrix}\begin{bmatrix}\cos\theta&\sin\theta\\0&1/\cos\theta\end{bmatrix}.$$ And for the $$3\times 3$$ case, the diagonal entries for $$U$$ should be something similar to $$\cos\theta_1\cos\theta_2, \cos\theta_3/\cos\theta_1, 1/\cos\theta_3\cos\theta_2.$$ Is there any previous work on these?

Recall that the Cholesky decomposition is a LU decomposition of a Hermitian matrix, where $$U$$ is the conjugate transpose of the lower-triangular matrix $$L$$. The analogue for an orthogonal matrix $$O$$ is $$O=PLR^{-1}$$ where $$P$$ is a permutation matrix, $$L$$ is lower triangular, and $$R$$ is such that $$PL=QR$$ with $$Q$$ orthogonal and $$R$$ upper-triangular.
So, up to a permutation $$P$$ of the columns of $$O$$, this orthogonal matrix is fully determined by an unconstrained lower-triangular matrix $$L$$ --- in this sence the "PLR-decomposition" is the analogue of the Cholesky decomposition.