Math 6420 • PDE

\begin{enumerate}
\item[(a)] Let
\[
  A=
  \begin{pmatrix}
    3 & 1\\
    1 & 3
  \end{pmatrix}.
\]
Find the maximum value of the quadratic form \(\langle Ax,x\rangle\) for \(\lVert x\rVert\le 1\).

\item[(b)] Prove that eigenvectors corresponding to distinct eigenvalues of a self-adjoint
matrix are orthogonal.

\item[(c)] Let \(H=-\frac{d^2}{dx^2}\) on \(L^2[0,1]\) with boundary conditions
\(\psi(0)=\psi(1)=0\). Find the eigenvalues and eigenvectors of \(H\). Show that the eigenvectors
of \(H\) corresponding to distinct eigenvalues are orthogonal.
\end{enumerate}