Gradient descent for an objective function I am reading the Wikipedia on gradient descent on a non-linear system just to get better understand of how they work under the hood. I can follow everything up until this line:

I understand the definition of a Jacobian, but what I don't understand is how this  equation above is true. Can someone point me to a resource, or maybe give me some intuition as to why this is true?
 A: By definitions:
\begin{equation}
\nabla F = \begin{bmatrix}
    \dfrac{\partial F}{\partial x_1} \\
    \vdots \\
    \dfrac{\partial F}{\partial x_n}\end{bmatrix},
\quad
G = \begin{bmatrix}
    g_1 \\
    \vdots \\
    g_m\end{bmatrix}
\end{equation}
\begin{equation}
J_G = \begin{bmatrix}
    \dfrac{\partial g_1}{\partial x_1} & \cdots & \dfrac{\partial g_1}{\partial x_n}\\
    \vdots & \ddots & \vdots\\
    \dfrac{\partial g_m}{\partial x_1} & \cdots & \dfrac{\partial g_m}{\partial x_n} \end{bmatrix}
\end{equation}
Now, we have:
\begin{equation}
J_G^\top\cdot G = \begin{bmatrix}
    \dfrac{\partial g_1}{\partial x_1} & \cdots & \dfrac{\partial g_m}{\partial x_1}\\
    \vdots & \ddots & \vdots\\
    \dfrac{\partial g_1}{\partial x_n} & \cdots & \dfrac{\partial g_m}{\partial x_n} \end{bmatrix}\cdot \begin{bmatrix}
    g_1 \\
    \vdots \\
    g_m\end{bmatrix} = \begin{bmatrix}
    g_1\dfrac{\partial g_1}{\partial x_1} + \cdots +  g_m\dfrac{\partial g_m}{\partial x_1}\\
    \vdots \\
    g_1\dfrac{\partial g_1}{\partial x_m} + \cdots +  g_m\dfrac{\partial g_m}{\partial x_m}\end{bmatrix}
\end{equation}
On the other hand, $i$-th row of $\nabla F$ is:
\begin{equation}
\dfrac{\partial F}{\partial x_i} = \frac{1}{2}\cdot\dfrac{\partial}{\partial x_i}(g_1^2 + \cdots + g_m^2) = g_1\dfrac{\partial g_1}{\partial x_i} + \cdots +  g_m\dfrac{\partial g_m}{\partial x_i},
\end{equation}
because 
\begin{equation}
F = \frac{1}{2}(g_1^2 + \cdots + g_m^2).
\end{equation}
and
\begin{equation}
\dfrac{\partial}{\partial x_i}(g_j^2) = 2g_j\dfrac{\partial g_j}{\partial x_i}.
\end{equation}
