I'm reading in a published work a generalization form of a messy equation and I couldn't get my head around it. Could you please share your thoughts.
Giving that $x$ and $y$ are complex vectors with the same dimensions: $$ y = \begin{bmatrix}y_1\\ y_2\\ \vdots \\ y_n \end{bmatrix}_{n \times 1} \qquad x = \begin{bmatrix}x_1\\ x_2\\ \vdots \\ x_n \end{bmatrix}_{n \times 1} $$ where $$y_1 = \operatorname{Re}(y_1)+j \operatorname{Im}(y_1), \qquad x_1 = \operatorname{Re}(x_1)+j \operatorname{Im}(x_1)$$
And $\lambda$ is a real numbers vector, $$ \lambda=\begin{bmatrix}\lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n \end{bmatrix}_{n \times 1} $$ and $\rho$ is a constant value.
Let us create a new vector called $\tilde{y}$ and stack the real and imaginary parts of the vector $y$ inside it, like this:
$$ \tilde{y} = \begin{bmatrix} \operatorname{Re}(y) \\ \operatorname{Im}(y) \end{bmatrix}_{2n \times 1} $$
now, an equation in this form:
$$ \begin{align} G&=\sum_n^n -\lambda_i^{\intercal} y_i + \frac{\rho}{2} ||x_i-y_i||^2_2 \\&= - \lambda_1 \: y_1\:+\frac{\rho}{2} \sqrt{\left(\begin{array}{c}x_1-y_1\end{array}\right)^{\intercal} \left(\begin{array}{c}x_1-y_1\end{array}\right)}\; - \lambda_2 \: y_2\:+\frac{\rho}{2} \sqrt{\left(\begin{array}{c}x_2-y_2\end{array}\right)^{\intercal} \left(\begin{array}{c}x_2-y_2\end{array}\right)} \cdots \cdots - \lambda_n \: y_n\:+\frac{\rho}{2} \sqrt{\left(\begin{array}{c}x_n-y_n\end{array}\right)^{\intercal} \left(\begin{array}{c}x_n-y_n\end{array}\right)} \end{align} $$
can be equivalently written as, $$ G = \frac{1}{2} \: \tilde{y} \: M \: \tilde{y} + c^{\intercal} \: \tilde{y} $$ Where $M$ is a positive diagonal matrix and $c$ is a real vector.
My question is, how did they arrive to the generalized equation and what will the contents of $M$ and $c$ be? The authors didn't elaborate on that!
Thank you.