Say we have a possibility complex $N\times N$ block matrix $M$ given by $$ M(t)= \begin{pmatrix}A&B(t)\\B(t)&C\end{pmatrix}, $$ where $A,B(t),C$ are $N/2\times N/2$ complex matrices and $B(t)=tB$, i.e. $B(t)$ is simply a matrix we can tune off to $0$ as we tune $t\rightarrow 0$. What can we say about the eigenvalues/eigenvectors of $M$ as a function of $t$?
I can see that at $t=0$ the eigenvalues and eigenvectors are those of $A$ and $C$ separately and I also expect some kind of continuity from $t=0$ to $t=1$, whatever that means.
I also see that at $t\rightarrow \infty$ one effectively has $$ M(\infty)= \begin{pmatrix}0&B(\infty)\\B(\infty)&0\end{pmatrix}, $$ which I am unfamiliar with this form.
Does anybody know anything about this problem? Perhaps references?