How to compute the eigenvalue condition number of a matrix

Question

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$

$$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$

I am a bit stuck on how to proceed solving this problem I know that the eigenvalue condition number equation is $κ(λ, A) = \|y\|\|x\|$. But other than that, I am lost.

Answer 1

The right and left eigenvector can be obtained by computing the kernel of $X_{\lambda}$ and $X_{\lambda}^*$, where $X_{\lambda}=A-\lambda I$, which in this case is $$ X_{4}=\begin{bmatrix}0 & 0 \\ 1000 & 2\end{bmatrix}. $$ So the right and left eigenvectors are given by $$ x=\begin{bmatrix}-2/1000\\1\end{bmatrix},\quad y=\begin{bmatrix}1\\0\end{bmatrix} $$ You can verify that $X_{\lambda}x=0$ and $y^*X_{\lambda}=0$. So (using the (more) correct definition), $$ \kappa_4\approx 500. $$

Answer 2

The condition number of an eigenvalue is actually $\frac{1}{y^*x}$ where x and y are right and left eigenvectors of your matrix with respect to the same eigenvalue. Before you apply the formula make sure you normalize your eigenvectors.