PCP Decomposition

If we have a real $2\times 2$ matrix $A$ with eigenvalue $\lambda =a-bi$ (where $b\ne 0$) and eigenvector $\vec{v}$, we can construct,

$$A=PC{P}^{-1}$$

Where

$$P=\left[\begin{array}{cc}\text{Re}(\vec{v})& \text{Im}(\vec{v})\end{array}\right]$$ $$C=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$$

$C$ is a rotation dilation matrix

IMPORTANT

We are looking at the eigenvalue $a-bi$ not $a\pm bi$