eig function MATLAB on complex symmetric matrix

Question

Suppose I have a complex symmetric matrix D (not Hermatian) which after diagonalization becomes E. Let, D = F + iG and E = H + iJ. E, H and J are diagonal matrices. I am using 'eig' function in MATLAB to diagonalize D, i.e., [V,E] = eig(D). I always find that if I apply 'eig' on matrices F and G, I always get H and J respectively. It is easy to understand if the matrix formed by V is always real, i.e., $\text{V}^{-1}\text{DV}=\text{V}^{-1}\text{FV}+\text{i}\text{V}^{-1}\text{GV}=\text{H}+\text{iJ}$. Why does it happen? Is there a theorem behind it? Or, it is just a shortcoming of the algorithm used in 'eig' in MATLAB?

Answer 1

$\mathbb{C}$ is an isomorphism to $R^2$.