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When I retrace the implementation of lmer from lme4 I faced a question regarding cholesky decomposition used for solving penalized least squares.

Consider a Cholesky decomposition of a matrix M with block design

$ M = \left( \begin{matrix} A & B\\ C & D \end{matrix} \right) = RR'$

And the Cholesky desomposition described in the paper:

$\left( \begin{matrix} \Lambda_\theta^TZ^TWZ\Lambda_\theta + I & \Lambda_\theta^TZ^TWX\\X^TWZ\Lambda_\theta & X^TWX \end{matrix} \right) = \left( \begin{matrix} L_\theta & 0\\R^T_{ZX} & R_X^T \end{matrix} \right) \left( \begin{matrix} L^T_\theta & R_{ZX}\\0&R_X \end{matrix}\right)$

How can I extract $L_\theta$ from this decomposition?

Let's say

$\Lambda_\theta^TZ^TWZ\Lambda_\theta + I \in \mathbb{R}^{q\times q}$

$\Lambda_\theta^TZ^TWX \in \mathbb{R}^{q \times p}$

$X^TWX\in \mathbb{R}^{p \times p}$

I guess then follows that

$L_\theta \in \mathbb{R}^{q\times q}$ ,$R^T_{ZX} \in \mathbb{R}^{p \times q}$ and $R_X^T \in \mathbb{R}^{p \times p}$

My Idea is that $L_\theta$ must be the upper left block matrix of $R$, $R^T_{ZX}$ must be the lower block matrix of $R$ and so on. Am I right with my assumptions?

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