Complex conjugation in scalar product?

[ATY]

Hey guys,
I got the following derivation for some physical stuff (the derivation itself is just math)
http://thesis.library.caltech.edu/5215/12/12appendixD.pdf
I understand everything until D.8.

After D.7 they get the eigenvalue and eigenvectors from ε. The text says that my δx(t) gets aligned in the same direction as the the eigenvektor of the largest eigenvalue (this would be my explanation for the fact that there is no eigenvector in the scalar product, but this might be wrong, cause I do not know much about eigenvectors).
But what I don't get is why in D.8. my δx(t) is suddenly complex conjugated. I can not find the reason for this.
I would be really happy about any explanation.

have a nice day
ATY

PS: sorry for the weird titel. Had no clue how to describe my problem (mea culpa)

 

[jvicens]

Dear ATY,

Thank you for your question. I have reviewed the derivation in the link you provided and I believe I can provide an explanation for the complex conjugation of δx(t) in D.8.

Firstly, let's review what we know from the previous steps. In D.7, the authors have obtained the eigenvalue equation εu = Au, where u is an eigenvector and A is a matrix. They then substitute u = δx(t)/|δx(t)|, where δx(t) is a vector representing the displacement of a particle at time t. This substitution makes sense because we are dealing with a physical system and the displacement vector should have a magnitude of 1.

Now, in D.8, the authors take the complex conjugate of the eigenvalue equation from D.7. This is done because the eigenvector u is complex, and in order to obtain a real solution for δx(t), we need to take the complex conjugate of the equation. This is a common step in solving eigenvalue problems in quantum mechanics.

In summary, the complex conjugation in D.8 is necessary in order to obtain a real solution for δx(t) and does not affect the physical interpretation of the problem.

I hope this explanation helps. If you have any further questions, please don't hesitate to ask.