Cfrgtkky

I am modeling an optimization problem using semi-definite programming. The optimization variable is a rank-1 matrix $X=xx^T$. The vector $x$ contains the power network voltages, which are complex values but they can be split in real and imaginary part. In literature, there are two ways to solve this; either implement the whole optimization problem is complex domain or in real domain. In complex domain, $x=[v_1 v_2,...,v_n]$ where $v_k$ is a complex value. In real domain, $x=[a_1 a_2,...,a_n,b_1,b_2,...,b_n]$ where $a_k$ and $b_k$ are real and imaginary components of $v_k$. Furthemore, $v_1 (a_1,b_1)$ are known and thus the corresponding elements in $X$ is set to these values.

The problem is when I implement the problem in complex domain; the solver gives me a rank-1 matrix solution (only 1 eigen value is non-zero), whereas in real domain, the solver gives me two eigen values and the sum of these two eigen values is exactly equal to eigen-value obtained from complex domain. That's strange for me. The value of objective function obtained from both problem also matches exactly. Theory says that I should receive 1 eigenvalue even in the case of real domain implementation.

So, can anyone shed some light that in the case of real domain, would I receive two eigenvalues? and if yes, why? and is there any relation between eigenvalues in real and complex domain?