Math 6410 • ODE

\textbf{Linear Systems.}
Consider the linear initial value problem
\[
\begin{aligned}
  \dot{x} &= Ax,\\
  x(0) &= x_0,
\end{aligned}
\]
where \(A\in\R^{n\times n}\) is constant, and \(x_0\in\R^n\) is given.
\begin{enumerate}
\item[(a)] Prove that the matrix exponential \(e^{At}\) exists and satisfies
\(\lVert e^{At}\rVert\le e^{\lVert A\rVert t}\), where \(\lVert\cdot\rVert\) is the operator
norm.
\item[(b)] Prove directly that \(x(t)=e^{At}x_0\) is the unique solution to the initial
value problem above.
\end{enumerate}