Why do antisymmetric eigenvalues have to be purely imaginary?

[SeReNiTy]

Why do they have to purely imaginary?

I got a proof that looks like Ax=ax
where a = eigenvalue

therefore Ax.x = ax.x = a|x|^2

Ax.x = x.(A^t)x
where A^t = transpose = -A
x.(-A)x = -b|x|^2

therefore a=-b, where b = conjugate of a

Now is this as far as i need to go?

 

[matt grime]

Write it as a=x+iy, b=x-iy, now you're saying what...