Math 6420 • PDE

Suppose that the square real matrix \(A=(a_{ij})\) satisfies \(a_{ij}=a_{ji}\) for all \(i,j\).
\begin{enumerate}
\item[(a)] Show that every eigenvalue \(\lambda\) of \(A\) must be real.

\item[(b)] Show that for any eigenpairs \((\lambda_1,u_1)\), \((\lambda_2,u_2)\), with
\(\lambda_1\ne\lambda_2\), the eigenvectors \(u_1\) and \(u_2\) must be orthogonal.

\item[(c)] Let
\[
  c=\min_{\lVert u\rVert=1}u^TAu.
\]
Prove that if \(c>0\), then \(A^{-1}\) exists. In this case, find an estimate for the matrix
operator norm \(\lVert A^{-1}\rVert\).
\end{enumerate}