I am currently reading Volume I of "Principles of Quantum Computation and Information" by Benenti-Casati-Strini. There is a section in chapter 3 which covers the decomposition of any arbitrary $2^n \times 2^n $ unitary matrix into simpler 2x2 matrices. One such scheme is the CSD decomposition, it involves decomposing the unitary matrix $U$ as follows:
\begin{align} U = \begin{bmatrix} L_0 & 0\\ 0 & L_1 \end{bmatrix} D \begin{bmatrix} R_0 & 0\\ 0 & R_1 \end{bmatrix}\end{align}
Where $L_0$, $L_1$, $R_0$, $R_1$ are themselves smaller unitary matrices, with half the dimensions of the original matrix $U$ (so $2^{n-1} \times 2^{n-1}$). And the matrix $D$ is another matrix defined as follows:
\begin{align} D = \begin{bmatrix} D_C & -D_S\\ D_S & D_C \end{bmatrix} \end{align}
Here the entries are once again $2^{n-1} \times 2^{n-1}$ matrices. In particular, they are diagonal matrices defined as follows:
\begin{align} D_C = \text{diag}(cos(\phi_1), cos(\phi_2), ... , cos(\phi_{2^{n-1}}),) \end{align}
\begin{align} D_S = \text{diag}(sin(\phi_1), sin(\phi_2), ... , sin(\phi_{2^{n-1}}),) \end{align}
What I am having trouble understanding is an example presented in the book for the special case of $n = 2$ qubits i.e the decomposition of a $4 \times 4 $ matrix in the following way using controlled $L_0$, $L_1$, $R_0$, $R_1$ operations:
<img src="https://i.ibb.co/60tNq52/CSDDecomp-Circuit.png" alt="The Circuit">
First of all, shouldn't the circuit elements be ordered in reverse, since if we matrix multiply the representation of $U$ mentioned above, we get:
\begin{align} U = \begin{bmatrix} L_0 D_C R_0 & -L_0D_SR_1\\ L_1D_SR_0 & L_1D_CR_1 \end{bmatrix} \end{align}
Where, for example, the operations in the the first row and first column are in the opposite order. The second question is, how do these break down into controlled gates applied only on the second qubit and the third and final question is, how would I go about generalizing this circuit to the case of a greater number of qubits.
