In this paper on diffraction grating efficiency, the author makes a point that I am not completely comprehending.
Equation 28) $H b_0 = M_{Q+1}b_{Q+1}+l + M'B$
It is then stated:
Under these conditions, the matrix $T$ of Eq. (20) has a size $4N + 2$, and we compute in each medium its $4N + 2$ eigenvectors $V_q^j$ of size $4N +2$. From the form of the $T$ matrix, it can be shown that the number of values of $r_q^0$ satisfying Eq. (27) is equal to 2N + 1. Thus the number of components of bo will be exactly equal to $2N + 1$. In the same way, the number of $r_j^{Q+l}$ satisfying $Im(r_j^{Q+4} ) > 0$, or $Im(r_j^{Q+l}) = 0$ and $Re(r_j^{Q+'}) > 0$, is equal to 2N + 1. Furthermore, as we will see in Section 5, the number of rjQ+I such that Im(rjQ+ ) > 0 is equal to 2N + I - P. So the number of components of bQ+' is equal to 2N + 1 - P. Thus, since B has P components, Eq. (28) is nothing but a linear system of 4N + 2 equations with 4N + 2 unknowns, which can be solved with a classical numerical method. This permits us to know b0 , bQ+l, and B. The values of bi in each medium can be deduced from Eq. (24).
I completely understand where he is coming from in terms of the sizes of the vectors, but I do not understand how the equation I posted can be used to solve $b_0 b_{Q+1} and$ $B$