I am using the complex eigenvalues for a low-rank approximation of the singular values for my matrix.
The algorithm I am using has an intermediate step of calculating the eigenvalues/vectors of a much smaller matrix.
In this step I actually have to take the square root of the diagonal matrix containing the eigenvalues.
But thank you very much, I will see if I can expand this into something that can be calculated using only real valued matrices.
EDIT: Later on I have figured out I am in fact calculating the eigenvalues for a PSD matrix, so the complex values are small enough to be ignored.