How to compute the eigenvalue condition number of a matrix 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. 

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),
$$
\kappa_4\approx 500.
$$
