Help with Matrix Equations

Question

So I am looking at this paper and I am lost on equation 28. Section 4A tells the size of all components, but it looks like none of the dimensions add up. So I guess I am asking how the author uses the sizes given to find that there are 4N+2 equations. Thanks

Answer 1

Take equation $20$ and replace the infinite vector $\mathbf \xi$ (that was of $N \rightarrow \infty$) with the finite vectors $\gamma_N$ from equation $18$ and $19$. These vectors have a dimension of $2(2N+1)$.

In equation $20$ the $m$-dimension of matrix $\mathbf T$ accords to the dimmensions of the corresponding vectors $\xi$ (it says "defined like vector $\xi$..."), however consequently, now when substituted by $\gamma_N$ the matrix $\mathbf T$ will follow and get the $m$-dimension of $(4N+2)$.

This is also in accordance with the matrix form (paragraph after equation $20$):

$$\begin{bmatrix} A & B \\[0.3em] C & D \\[0.3em] \end{bmatrix}$$

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With regard to equation $28$ following your question:

$$\underbrace{\mathbf {b^0}}_{4N+2\rightarrow 2N+1} = \dots+ \underbrace{M'}_{(4N+2)\times P} \underbrace{\mathbf B}_{P}$$

what I understand is that despite of $M'$s dimmension because of the conditions of inequalities $27$, iff $r_q^0$ satisfying $27$ a row would remains, all other rows will cancel so you will have only $2N+1$ equations left.

This is how I understand the author yet and hope make physically sense.