Math 6420 • PDE

Let
\[
  A=
  \begin{pmatrix}
    2 & 1 & 1\\
    1 & 2 & 1\\
    1 & 1 & 2
  \end{pmatrix}.
\]
\begin{enumerate}
\item[(a)] Find the eigenvalues and eigenvectors of \(A\), and the range of the function
\[
  \phi(x)=\frac{\langle Ax,x\rangle}{\langle x,x\rangle}
\]
where \(x=(x_1,x_2,x_3)\) is a real-valued vector.

\item[(b)] Compute \(\exp(A)\).

\item[(c)] Consider the equation
\[
  Ax=\mu x
\]
Find all solutions for \(\mu=1\) and for \(\mu=3\).
\end{enumerate}