We need to solve the following complex least square problem:\begin{align*}\min\limits_X&&\|Y-X\|^2\\\text{s.t.}&&X^TAX=0\end{align*}where the complex matrix $Y$ and the real matrix $A$ are given.
My Thoughts:
Let $L=\textbf{Tr}\{YY^H-YX^H-XY^H+XX^H\}+\lambda(X^TAX)$ be the Lagrangian function. Thus, we have
\begin{equation} \frac{\partial L}{\partial X}=0 \Longrightarrow -Y^* + X^* + \lambda(A+A^T)X=0 \end{equation}
\begin{equation} \frac{\partial L}{\partial X^*}=0 \Longrightarrow -Y +X =0 \end{equation}
So we have $X=Y$. Did we do something wrong? Can anyone help us? Thanks a lot!