I am trying to calculate the characteristic polynomial of a block tridiagonal matrix and I need some help. This matrix is a representation of a tight-binding Hamiltonian of a finite grid of graphene, it'll be really cool to get the analytical eigenenergies of the system to improve the numerical methods.
$t_1$ is the intra cell hopping, $t_2$ is the inter hopping along x direction and $t_3$ is the inter hopping along y direction. The structure of the Hamiltonian is as follows:
$$M=\begin{bmatrix} \Delta & T & 0 & \cdots & 0 \\ T^t & \Delta & T & \cdots & 0 \\ 0 & T^t & \Delta & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & T^t & \Delta \end{bmatrix}$$
where:
- $\Delta$ is a $n \times n$ tridiagonal matrix with zeros on the main diagonal, and the upper and lower diagonals alternate with $t_1$ and $t_2$, starting with $t_1$.
$$\Delta = \begin{bmatrix} 0 & t_1 & 0 & \cdots & 0\\ t_1 & 0 & t_2 & \cdots & 0\\ 0 & t_2 & 0 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & t_1 & 0 \end{bmatrix}$$
- $T$ is a $n \times n$ matrix with zeros on the main diagonal, and the first lower diagonal is all $t_3$.
$$T = \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 \\ t_3 & 0 & 0 & \cdots & 0 \\ 0 & t_3 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & t_3 & 0 \end{bmatrix}$$
- $T^t$ is simply the transpose of $T$.
I am looking for an effective way to calculate the characteristic polynomial of this block tridiagonal matrix. Any suggestions or references would be greatly appreciated.
I already know how to calculate the characteristic polynomial of $\Delta$ and obviously $T$ if it helps.