Zeeman Effect with a matrix involved

[M. next]

Please refer to picture (1). This the equation of motion for an electron in a magnetic field along the z direction and moving with a certain velocity. Where Ω is the Larmor's frequency.
Then refer to picture (2), that is the matrix representing the previous equation. How was this matrix arranged. I didn't understand a thing.

Any help is much appreciated.
Please excuse me for the bad resolution of the images.

 

[daveyrocket]

I think this is done by assuming that each component of r is of the form exp(i*omega*x). Then you put that in the first equation and work out what the equations are for each component and rewrite it in matrix form.

 

[M. next]

Yes, that's true, your assumption is written also.. Can you please elaborate more?

 

[daveyrocket]

Do you have a more specific question? You should be able to work it out yourself to get the result.