# Matrix perturbation and eigenvector

$$Ae=\lambda e$$

$$e^Te=1$$

$$A$$ is a real matrix. $$\lambda, e$$ are real.

$$(A+\Delta A)(e+\Delta e)=(\lambda+\Delta \lambda)(e+\Delta e)$$

Neglecting small terms,

$$\Delta Ae +A \Delta e=\Delta \lambda e+\Delta e \lambda \tag{1}$$

$$e^T \times (1)$$

$$e^T\Delta Ae +e^TA\Delta e=e^T\Delta \lambda e+e^T\Delta e \lambda$$

$$e^T\Delta Ae +\lambda e^T\Delta e=e^T\Delta \lambda e+e^T\Delta e \lambda$$

$$e^T\Delta Ae =e^T\Delta \lambda e$$

$$\Delta \lambda=e^T\Delta Ae \tag{2}$$

Is this correct?

How to derive for $$\Delta e?$$

I have seen somewhere that the eigenvector is almost unchanged for small perturbation in a matrix. How to prove it?

I am looking to prove that

when $$\Delta A$$ is small, $$\Delta e \rightarrow 0$$

Update : This thread has an answer, but does it solve my problem? What happens when eigenvalues are same?