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__primal__

min <C,X>

s.t. <A_i,X> = b_i, i=1,...,m

X>=0

__dual__

max <b,y>

s.t. y_1*A_1 + ... + y_m*A_m <=C

where the data A_i and C are assumed to be real symmetric matrices in many textbooks and online materials.

If we consider complex SDP where A_i and C are Hermitian, will all the results about real SDP be correct by replacing the real matrices to Hermitian matrices? To be precious, will the primal and dual forms be still the same? Do the interior-point methods for real SDP work for complex SDP?