(revised)-Eigenvalues and eigenvectors of a special tridiagonal matrix (nonsymmetric)?

Question

I need to consider to find the eigenvalues and eigenvectors of the following tridiagonal matrix, \begin{equation} \begin{bmatrix} \gamma & \beta \\ \alpha & \gamma & \beta \\ & \ddots & \ddots & \ddots \\ & & \alpha & \gamma & \beta \\ & & & -1 & 1 \\ \end{bmatrix}_{m\times m},\quad~~m\in\mathcal{N}^{+}, \end{equation} where $\gamma\in\mathbb{C}$, $\alpha = -\frac{1 + \gamma}{2}$ and $\beta = \frac{1 - \gamma}{2}$. I want to find some analytical form of these eiganvalues and eigenvector. But I have no idea about such problem. I remember that if $A(m,m-1) = \alpha,~A(m,m) = \gamma$, the problem will be easy to find the answer. Are there some ideas about this problem?