Let $H$ be a square complex symmetric (not Hermitian) matrix such that $H_{11}$ (the entry at the top left corner) is very large compared to any other entries. Assume $H_{11}$ is real. When can it guaranteed that $H$ has a real eigenvalue?
Using Gershgorin's circle theorem I can prove that there exists an eigenvalue that is near to $H_{11}$, but I can't guarantee that it must real.
I'm being vague on purpose about $H_{11}$ being 'very large' because I'm interested in general conditions under which the above can be stated. Any ideas appreciated.
Answering my own question. A slight strengthening of Gershgorin's circle theorem says that if $k$ of the circles are disjoint from the other $n-k$ then they must contain exactly $k$ eigenvalues. Furthermore, any circle that is centered at a real value must contain either only real or conjugate eigenvalues (because two eigenvalues that are conjugate to each other are the same distance away from any point on the real line).
So a sufficient condition on $H$, since $H_{11}$ is real, is that its eigenvalues are either real or come in conjugate pairs, AND that the corresponding circle is isolated from the rest. This guarantees there is exactly one real eigenvalue inside of it.