Let's say I have a diagonalizable $3\times3$ matrix $$ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$
with 3 distinct eigenvalues $\lambda_1, \lambda_2, \lambda_3$. Let's say I now add a small perturbation to $i$ of the form $-k^2$ so that the matrix becomes:
$$ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i - k^2 \end{pmatrix}$$
Is there are formula for how the eigenvalues will change? I'm not even sure where to start.
Write $A = \begin{bmatrix}a & b & c \\ d & e & f \\ g& h & i\end{bmatrix}.$
Say that $A$ is Hermitian that is $A = \overline{A}^T$ then you could use Weyl's perturbation theorem which says that if $B$ is another Hermitian matrix then
$$\max_{1\leq i \leq 3}|\lambda_i(A+B)-\lambda_i(A)|\leq \|B\|$$
Here $\lambda_i$ is the map which takes Hermitian matrices to their eigenvalues ordered non-increasingly.
In your case
$$B = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & k^2\end{bmatrix}$$