Complex function and Jacobian matrix Given some complex-differentiable function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined $f(x,y)=u(x,y)+iv(x,y)$, we know the Cauchy-Riemann equations hold, so:
$$\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}\quad\textrm{and}\quad\dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}$$
Then, we can write the Jacobian for the function:
$$\begin{bmatrix}\dfrac{\partial u}{\partial x}&\dfrac{\partial u}{\partial y}\\-\dfrac{\partial u}{\partial y}&\dfrac{\partial u}{\partial x}\end{bmatrix}$$
At this point, my textbook claims that this matrix has the same effect on $\mathbb{C}$ as multiplication by the complex number $a=\dfrac{\partial u}{\partial x}-i\dfrac{\partial u}{\partial y}$ (therefore, $a$ is the derivative of $f$), but I'm having a hard time seeing why that's the case, and how this value of $a$ was reached in the first place. Any suggestions?

$$\begin{pmatrix}
\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\
-\frac{\partial u}{\partial y}&\frac{\partial u}{\partial x}\\
\end{pmatrix}\begin{pmatrix}
x\\y\\
\end{pmatrix}
=\begin{pmatrix}
\frac{\partial u}{\partial x}x +\frac{\partial u}{\partial y}y\\
-\frac{\partial u}{\partial y}x+\frac{\partial u}{\partial x}y\\
\end{pmatrix}$$
On the other hand, 
$$\left(\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}\right) (x+iy)=
\frac{\partial u}{\partial x}x +\frac{\partial u}{\partial y}y+
\left(-\frac{\partial u}{\partial y}x+\frac{\partial u}{\partial x} y\right)i$$
Compare  the terms and see they are the same.
