Explicit Scheme, final stability analysis
The eigenvalues of \( \mathbf{A} \) are \( \lambda_i=1-\alpha\mu_i \), with \( \mu_i \) being the
eigenvalues of \( \hat{B} \). To find \( \mu_i \) we note that the matrix elements of \( \hat{B} \) are
$$
\begin{equation*}
b_{ij} = 2\delta_{ij}-\delta_{i+1j}-\delta_{i-1j},
\end{equation*}
$$
meaning that we
have the following set of eigenequations for component \( i \)
$$
\begin{equation*}
(\hat{B}\hat{x})_i = \mu_ix_i,
\end{equation*}
$$
resulting in
$$
\begin{equation*}
(\hat{B}\hat{x})_i=\sum_{j=1}^n\left(2\delta_{ij}-\delta_{i+1j}-\delta_{i-1j}\right)x_j =
2x_i-x_{i+1}-x_{i-1}=\mu_ix_i.
\end{equation*}
$$