Math 6610 • Analysis of Numerical Methods I

{Eigenvalue Problems}
Let \(A\) be a real \(n\times n\) matrix with simple eigenvalue \(\lambda\), right
eigenvector \(x\) (i.e., \(Ax=\lambda x\)), and left eigenvector \(y\) (i.e.,
\(y^*A=\lambda y^*\)). Assume that \(\lVert x\rVert_2=\lVert y\rVert_2=1\).
\textbf{a)} Show that the condition number for \(\lambda\) to changes in \(A\) is given
by \(s(\lambda)=1/\lvert y^*x\rvert\). \textbf{b)} Give an example of a matrix \(A\)
which has a badly conditioned eigenvalue, and another matrix \(A\) which has only
well-conditioned eigenvalues. \textbf{c)} Sketch a reasonably efficient algorithm
for finding the largest eigenvalue of a symmetric matrix \(A\) with distinct
eigenvalues. How fast do you expect this algorithm to converge?