Leading eigenvalue of a strange tridiagonal matrix with matrix sub-blocks

Question

Let's assume the following crazy matrix

\begin{equation} P = \begin{pmatrix} \mathbb{0}_{1\times 1} & \alpha \mathbb{1}_{1\times N} & \mathbb{0}_{1\times\frac{N(N-7)}{2!}} & \mathbb{0}_{1 \times \frac{N(N-10)(N-11)}{3!}} & \ldots \\ \alpha \left(\mathbb{1}_{1\times N}\right)^{\intercal} & \mathbb{0}_{N\times N} & \mathbb{G}_{N\times \frac{N(N-7)}{2!}} & \mathbb{0}_{N \times \frac{N(N-10)(N-11)}{3!}} & \ldots \\ \mathbb{0}_{\frac{N(N-7)}{2!}\times 1} & (\mathbb{G}_{N\times \frac{N(N-7)}{2!}})^{\intercal} & \mathbb{0}_{\frac{N(N-7)}{2!} \times \frac{N(N-7)}{2!}} & \mathbb{F}_{\frac{N(N-7)}{2!} \times \frac{N(N-10)(N-11)}{3!}} & \ldots \\ \mathbb{0}_{\frac{N(N-10)(N-11)}{3!} \times 1} & \mathbb{0}_{\frac{N(N-10)(N-11)}{3!} \times N} & (\mathbb{F}_{\frac{N(N-7)}{2!} \times \frac{N(N-10)(N-11)}{3!}})^{\intercal} & \mathbb{0}_{\frac{N(N-10)(N-11)}{3!} \times \frac{N(N-10)(N-11)}{3!}} & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{pmatrix} \end{equation}

This matrix is of the form of a tridiagonal matrix \begin{equation} \begin{pmatrix} 0 & A & 0 & 0 & \ldots \\ A & 0 & B & 0 & \ldots \\ 0 & B & 0 & C & \ldots \\ 0 & 0 & C & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{pmatrix} \end{equation} with zero diagonal elements.

Some differences:

1. The elements of $P$ are matrices themselves.
2. The matrix blocks are not square, neither symmetric. Some (maybe?) useful properties of these matrices will be given below (together with the example).
3. The matrices $\mathbb{G}, \mathbb{F}, \ldots$ are sparse. All their nonzero elements are equal to $\alpha$, similar to $(2, 1)$ and $(1, 2)$ elements which are vectors of $N$ elements equal to $\alpha$.
4. $N$ is a variable with $N \in \mathbb{N}$ and for our case $N>9$. $N$ is never a prime number. There is always a decomposition $N = N_x \times Ny$.
5. The number of blocks of the matrix $P$ are equal to $Z $, where Z is found from the following construction (we concentrate on the case $Nx = Ny$): \begin{align} (Nx, Ny) &\rightarrow Z \\ (3, 3) &\rightarrow 3 \\ (4, 4) &\rightarrow 4 \\ (5, 5) &\rightarrow 5 \\ (6, 6) &\rightarrow 6 + 6 = 12\\ (7, 7) &\rightarrow 7 + 7 = 14 \\ (8, 8) &\rightarrow 8 + 8 = 16 \\ (9, 9) &\rightarrow 9 + 9 + 9 = 27 \\ (10, 10) &\rightarrow 10 + 10 + 10 = 30 \end{align} and so on.
6. For the case of $(Nx, Ny) = (3, 3)$, the three nonzero blocks of the matrix P are $\alpha \mathbb{1}, \mathbb{G}, \mathbb{F}$, with the explicit form: \begin{equation} \alpha \mathbb{1}_{1 \times N} = \begin{pmatrix} \alpha & \alpha & \alpha & \alpha & \alpha & \alpha & \alpha & \alpha & \alpha \end{pmatrix}, \end{equation} \begin{equation} \mathbb{G}_{9\times 9} = \begin{pmatrix} 0 & 0 & \alpha & 0 & 0 & 0 & \alpha & 0 & 0 \\ \alpha & 0 & 0 & 0 & 0 & 0 & 0 & \alpha & 0 \\ 0 & \alpha & 0 & 0 & 0 & 0 & 0 & 0 & \alpha \\ \alpha & 0 & 0 & 0 & 0 & \alpha & 0 & 0 & 0 \\ 0 & \alpha & 0 & \alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \alpha & 0 & \alpha & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \alpha & 0 & 0 & 0 & 0 & \alpha \\ 0 & 0 & 0 & 0 & \alpha & 0 & \alpha & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \alpha & 0 & \alpha & 0 \\ \end{pmatrix}, \end{equation} and \begin{equation} \mathbb{F}_{9\times 3} = \begin{pmatrix} 0 & \alpha & 0 \\ \alpha & 0 & 0 \\ 0 & 0 & \alpha \\ \alpha & 0 & 0 \\ 0 & 0 & \alpha \\ 0 & \alpha & 0 \\ 0 & 0 & \alpha \\ 0 & \alpha & 0 \\ \alpha & 0 & 0 \\ \end{pmatrix}, \end{equation} where $\alpha$ is real.
7. A way to think about the nonzero submatrix blocks is as an adjacency matrix of a k-regular p-XORSAT instance (in reality not random at all, but a very specific one. I don't know if this is any relevant...). For the 3rd column it is a 3-regular 2-XORSAT, for the 4th, a 3-regular 1-XORSAT (trivial). For bigger matrices, $P$, similar results follow. In our case, the matrices have elements $\{0, \alpha\}$ instead of the $\{0, 1\}$ of the standard form of an adjacency matrix.
8. The matrix comes up in a first order perturbation theory of a quantum spin model hamiltonian.

Goals (most important first)

1. Show that the leading eigenvalue is always unique.
2. Is there a possibility to find a general form for the eigenvalues of this matrix? Disclaimer: I know pretty well that I have not explained the way the $\mathbb{G}, \mathbb{F}, \ldots$ matrices are obtained. This is a long process by itself. Before doing it (if there is interest), I would like to know whether there are any known results or whether any procedure exists for eigenvalues for these general, nonsquare type blocks.

I am able to show that the leading eigenvalue is unique for the matrix $P$ only on a case by case basis and only numerically. Proving it in a general case would be a dream. Any ideas would be really appreciated!