The case is this:

Imagine I have an electron polarised in the positive z-direction (call it |z_up> ). And can prepare this for instance with a Stern Gerlach apparatus.

Now I perform a measurement on the spin in an orthogonal direction, say the x-direction. I either find |x_up> or |x_down>. Whatever the result, it can be expressed as a superposition of |z_up> and |z_down>, more specifically as one of the following 2 vectors: (1,1) or (1,-1) in the z-basis.

Until this point, I think I got it right. Now comes what disturbs me:

Measuring is mathematically represented by a Hermitian matrix, in this case the matrix S_x. But letting S_x act on (1,0) (wich is |z_up>) gives me (0,1), wich would say that my spin flipped in the z-direction.

Now I think the problem is that I'm misusing/interpreting S_x...but I can't figure out what would be the correct procedure...