\(4.2^{\star} \quad\) Consider the solution of \(y^{\prime}=\Lambda y\) where
$$
\Lambda=\left[\begin{array}{ll}
\lambda & 1 \\
0 & \lambda
\end{array}\right], \quad \lambda \in \mathbb{C}^{-} .
$$ a Prove that
$$
\Lambda^{n}=\left[\begin{array}{cc}
\lambda^{n} & n \lambda^{n-1} \\
0 & \lambda^{n}
\end{array}\right], \quad n=0,1, \ldots
$$
b Let \(g\) be an arbitrary function that is analytic about the origin. The \(2
\times 2\) matrix \(g(\Lambda)\) can be defined by substituting powers of
\(\Lambda\) into the Taylor expansion of \(g\). Prove that
$$
g(t \Lambda)=\left[\begin{array}{cc}
g(t \lambda) & \operatorname{tg}^{\prime}(t \lambda) \\
0 & g(t \lambda)
\end{array}\right]
$$
c By letting \(g(z)=e^{z}\) prove that \(\lim _{t \rightarrow \infty}
y(t)=\mathbf{0}\).
d Suppose that \(y^{\prime}=\Lambda y\) is solved with a Runge-Kutta method,
using a constant step size \(h>0\). Let \(r\) be the function from Lemma 4.1.
Letting \(g=r\), obtain the explicit form of \([r(h \Lambda)]^{n}, n=0,1, \ldots\)
e Prove that if \(h \lambda \in \mathcal{D}\), where \(\mathcal{D}\) is the linear
stability domain of the RungeKutta method, then \(\lim _{n \rightarrow \infty}
y_{n}=\mathbf{0}\).
