I want to solve the following constrained optimization problem $$ \arg\min_X \| Y - X \|_2^2 , \quad Y,X \in \mathbb{C}^{N \times 1}$$ $$ s.t. A X = 0 , \quad A \in \mathbb{C}^{N-2 \times N}$$
If I reduce the problem to the real case, I manage to find the solution using the Lagrangian : $$ X= (I-A^H(AA^H)^{-1}A)Y $$
However, in the complex case, I can't get any solutions. Any help would be appreciated! Thanks in advance!