Math 6420 • PDE

Suppose that \(A\) is an \(n\times n\) matrix with \(n\) distinct eigenvalues.
\begin{enumerate}
\item[(a)] How are the eigenvalues of \(A\) and \(A^*\) related?
\item[(b)] Show that the eigenvectors of \(A\) and \(A^*\) form a biorthogonal set, that is, if
\(\{\phi_i\}_{i=1}^n\) and \(\{\psi_i\}_{i=1}^n\) are the eigenvectors of \(A\) and \(A^*\),
appropriately ordered, then \(\langle\phi_i,\psi_j\rangle=0\) if \(i\ne j\).
\item[(c)] Establish that \(\langle\phi_i,\psi_i\rangle\ne0\).
\end{enumerate}