# Eigenvalue response to perturbation

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.

• Are there any conditions on you $3\times 3$ matrix? – OgvRubin Oct 25 '18 at 21:15
• @OlofRubin it is diagonalizable, has 3 distinct eigenvalues – Mike Flynn Oct 25 '18 at 21:17
• Yes, there is a formula, i.e., a cubic equation (characteristic polynomial) and its roots. – Dietrich Burde Oct 25 '18 at 21:19

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.
$$B = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & k^2\end{bmatrix}$$