# Cholesky Decomposition (in lmer from lme4)

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?