First let me put some context to my question. It is not hard to find a derivation of optimal solution of
\begin{eqnarray} \text{Minimize } & \phi = \frac{1}{2}\left\|Ax - y\right\|^{2}_{2} \nonumber \end{eqnarray} in least square sense ($x=(A^{T}A)^{-1}A^{T}y$).
We also can do it by Lagrange multiplier:
\begin{eqnarray} \text{Minimize } & x^{T}x \nonumber\\ \text{Subject to } & Ax = y \nonumber \end{eqnarray} Introduce Lagrange multipliers: $L(x,\lambda) = x^{T}x + \lambda^{T}(Ax-y)$
Optimality conditions are \begin{eqnarray} \nabla_{x}L &=& 2x + A^{T}\lambda = 0, \\ \nabla_{\lambda}L &=& Ax-y = 0. \end{eqnarray}
From the first condition, $x = -A^{T}\lambda/2$
Substitute into second to get $\lambda = -2(AA^{T})^{-1}y$
Hence $x=A^{T}(AA^{T})^{-1}y\equiv(A^{T}A)^{-1}A^{T}y$
MY QUESTION:
I want do the same derivation with Lagrange multipliers, but now I want consider complexs numbers. So that:
\begin{eqnarray} \text{Minimize } & x^{H}:x \nonumber\\ \text{Subject to } & Ax = y \nonumber \end{eqnarray} where $H$ is the Hermitian.
[(:) Should be a Frobenius inner product?]
How can I do it?