Let $H$ be a square matrix such that its top left entry, namely $H_{11}$, is very large (in absolute value) compared to the rest of the entries. It is my intuition that $H$ must have an eigenvalue that is very close to $H_{11}$. Under what conditions can this be guaranteed?
I pose this question with no restrictions on the entries of $H$, to explore what can be said in different cases. In my particular case, I am interested in the case when $H$ is complex symmetric (not Hermitian).
I have tried using the Gershgorin circle theorem, but it cannot guarantee that the circle corresponding to $H_{11}$ is non-empty. (The theorem guarantees that each eigenvalue must lie within at least one circle, but not that each circle contains at least one eigenvalue.)
A strengthened version of the Gershgorin circle theorem does guarantee that if the circles are grouped into disjoint clusters, the number of eigenvalues in each cluster are equal to the number of disks there. The proof is as follows: let $A=(a_{ij})$, $D=diag(a_{ii})$ and $A=D+E$. Define $A_t=D+tE$ and imagine increasing $t$ from 0 to 1, keeping track of the Gershgorin disks at every moment. $A_0=D$ so the disks are points and each eigenvalue is at each point; increasing $t$ increases the radii of the disks proportionally. Try to convince yourself that each eigenvalue of $A_t$, as a root of a monic polynomial with coefficients continuous in $t$, is also a continuous function of $t$ (especially considering multiplicities). Then each eigenvalue must stay in each expanding disk, until perhaps some of the disks join, in which case the eigenvalues are free to move around inside the cluster but can NOT jump outside to another cluster. Thus we have the desired result at $t=1$: each cluster of $k$ disks must contain $k$ eigenvalues, counting multiplicity.
Finally, for your question, since the disk centered at $H_{11}$ will be very far away in $\mathbb{C}$ from the other disks (and have a radius much smaller than the distance), it will have to contain exactly one eigenvalue, the one that started as $H_{11}$ itself when $t=0$.