# Intuition on a complex vector

1. Oct 26, 2012

### Shing

Would anyone be kind enough shed some light on the physics Intuition of a vector in a linear space over a complex field for me?

Furthermore, what does z^z mean?

2. Oct 26, 2012

### mathwonk

a complex structure on a real vector space is just a linear operator J such that J^2 = -Id. It is usually thought iof as a 90degree rotation counterclockwise. but you have tod ecide what plane to rotate in.

so whereas a real vector is thought of as an arrow, i.e. having both length and direction, a complex vector also has an associated perpendicular direction.

If you think of an arrow as a real arrow that you shoot, notice it has feathers and if it has a traditional native american arrowhead, it also has a sharp flat blade. That blade determines a 2 dimensional plane containing the arrow. If that blade is also painted half red and half blue, then you can think of the blue side as determining the counterclockwise direction in that plane.

so that allows you to rotate the arrow 90 degrees in the plane of the blade, and towards the blue half of the arrowhead.

I don't know where in physics this concept comes up naturally, (maybe particles with "spin"?) but that is the data

that it determines.

3. Nov 1, 2012

### Shing

I am sorry that I can't understand it completely (since English is my 2nd language)
so my understanding is:
A vector with complex structure can be thought of a arrow(vector itself) with feathers (the complex number stays with the vector). While arrow pointing position, the feather itself no meaning of position but will affect the direction of the arrow.
(and that is indeed a brilliant idea!)

I would like to discuss a further question:
since both 2-D vector and a complex number are kinds of ordered pair, (but with different operations), if given another set of operation, I can create another "whatever" numbers too?
so the spirit of algebra is really about operation?