I have a eigenvector from some matrix.
$$(i \sin(\phi), 1+i \cos(\phi))$$ where $$ 0\le\phi\le\pi $$ and $i=(\sqrt-1)$
I tried to draw the locus of these points on the complex plane.(it should be a branch of hyperbola)
Once I let $x=i\sin(\phi), y= 1+i\cos(\phi)$ , I could get hyperbolic curve. $$y=1+i(1+x^2)^\frac{1}{2}$$
However, the problem is that $x$ is not real.
I wonder whether this is okay.