= Square-root splitting of a defective double eigenvalue
{title2=$\lambda_\pm=(d\pm\sqrt{d^2+4\epsilon})/2$}
The model <matrix> $B=\begin{pmatrix}0&1\\\epsilon&d\end{pmatrix}$ has a defective double <eigenvalue> when $d=\epsilon=0$. Its <characteristic polynomial> is $\lambda^2-d\lambda-\epsilon$, giving the displayed roots. For fixed nonzero $d$ the small root is $-\epsilon/d+O(\epsilon^2)$ and the other is $d+\epsilon/d+O(\epsilon^2)$. These expansions cease to be ordered at $d=O(\sqrt\epsilon)$, the <distinguished limit> in which both roots split on a square-root scale. This is the generic leading behavior when a perturbation closes a two-dimensional <Jordan block> through a nonzero lower-left entry; it need not occur for a perturbation that leaves that entry zero.
Back to article page
