Math 6410 • ODE

Let \(A\) be a constant \(n\times n\) matrix, and define the matrix
exponential by
\[
e^A=\sum_{k=0}^{\infty}A^k/k!.
\]

\begin{enumerate}
\item[(a)] Prove that \(e^{At}\) is differentiable (you may use known theorems about
scalar power series), and that it satisfies the initial value problem \(X'=AX\),
\(X(0)=I\) (the identity matrix). Justify your steps.

\item[(b)] Let \(A\) have distinct eigenvalues. Prove that for the origin to be a
globally asymptotically stable point for \(X'=AX\) it is sufficient that all of the
eigenvalues of \(A\) have negative real parts.

\item[(c)] Is the condition in b) that \(A\) have distinct eigenvalues required in
order to prove the result? (no proof necessary)

\item[(d)] Is the sufficient condition in b) also necessary? Justify your answer.
\end{enumerate}

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\textit{DE Qualifying Examination 1992}