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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?

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Here is one study of the analogue of the Cholesky decomposition for orthogonal matrices: Unconstrained representation of orthogonal matrices with application to common principle components (2019).
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.

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  • $\begingroup$ Thanks. Any relationship with the Euler angles? $\endgroup$ – Jiyuan Zhang Jan 16 at 13:25
  • $\begingroup$ none that I know of. $\endgroup$ – Carlo Beenakker Jan 16 at 14:28

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