Explicit Scheme, eigenvalues and stability
We can obtain closed-form expressions for the eigenvalues of \( \mathbf{A} \). To achieve this it is convenient
to rewrite the matrix as
$$
\begin{equation*}
\mathbf{A}=\hat{I}-\alpha\hat{B},
\end{equation*}
$$
with
$$
\begin{equation*}
\hat{B} =\begin{bmatrix}2&-1&0& 0 &\dots\\
-1&2&-1& 0&\dots \\
\dots & \dots & \dots & \dots & -1 \\
0 & 0 &\dots &-1&2\end{bmatrix}.
\end{equation*}
$$