Proving that multiplication acts linearly on the complex plane as the composition of a dilation and a rotation.

Question

Primary purpose of this problem: To prove that the multiplication by $z_0 \neq 0$ acts linearly on the complex plane as the composition of a dilation and a rotation.

I have been asked to prove that there exists $r_0>0$ and $\theta_0 \in \mathbb{R}$ such that the matrix $A_{z{_0}}$ can be decomposed as:

$$A_{z{_0}}=\begin {bmatrix} r_0 & 0 \\ 0 & r_0 \end{bmatrix} \begin {bmatrix}\cos(\theta_0) & -\sin(\theta_0)\\ \sin(\theta_0) & \cos(\theta_0) \end{bmatrix} =: D_{r_{0}}R_{\theta_{0}}$$ where $D_{r_{0}}$ represents dilation and $R_{\theta_{0}}$ represents rotation.

I have the basic intuition regarding the concepts in this problem. The idea of multiplying these two matrices allows us to get combination of Dilation and Rotation, however, I am not sure exactly how to "prove" this.

A few relations that I have come up with:

1. The matrix expansion of $A_{z_0}=\begin {bmatrix} r\cos(\theta_0) -r\sin(\theta_0)\\ r\sin(\theta_0) + r\cos(\theta_0) \end{bmatrix}$
2. Let $z=x+iy$. In polar, this is represented by $z=re^{i\theta}=r \left(\cos(\theta)+i(\sin(\theta)\right)$. I was also able to identify that the rotation matrix comes from: $\cos(\theta)+i(\sin(\theta))$
3. If we use Matrices to represent Euler: $z=re^{i\theta}=\left(\cos(\theta)+i(\sin(\theta)\right)$, we will have $$z=\begin {bmatrix} \cos(\theta_0) & -\sin(\theta_0)\\ \sin(\theta_0) & \cos(\theta_0) \end{bmatrix} \begin {bmatrix} r_0 \\ r_0 \end{bmatrix}=\begin {bmatrix} r\cos(\theta_0) -r\sin(\theta_0)\\ r\sin(\theta_0) + r\cos(\theta_0) \end{bmatrix}$$

However, I need some help understanding how to actually prove what the $r_0>0$ and $\theta_0 \in \mathbb{R}$ is.

Answer 1

If $z_0\in\Bbb C\setminus\{0\}$, then $z=\rho_0\bigl(\cos(\theta_0)+i\sin(\theta_0)\bigr)$, for some $\rho_0>0$ and some $\theta_0\in\Bbb R$. So, if $z\in\Bbb C$, and if you write $z$ as $a+bi$ ($a,b\in\Bbb R$), then$$z_0\times z=\rho_0\bigl(\cos(\theta_0)a-\sin(\theta_0)b+\bigl(\cos(\theta_0)b+\sin(\theta_0)a\bigr)i\bigr)$$and therefore multiplication by $z_0$ the composition of $w\mapsto \rho_0 w$ (a dilation) with the rotation whose matrix with respect to the basis $\{1,i\}$ is $\left[\begin{smallmatrix}\cos(\theta_0)&-\sin(\theta_0)\\\sin(\theta_0)&\cos(\theta_0)\end{smallmatrix}\right]$.