Let $A \in \mathbb{R}^{n \times n}$ be a real matrix. suppose $\lambda = \alpha + i \beta$ is a complex eigenvalue of $A$ with complex eigenvector $w = u+ i v$ in $\mathbb{R}^{n}$. Let $\bar{\lambda } = \alpha - i \beta$ and $\bar{w} = u- i v$ be the complex conjugates of $\lambda$ and $w$. Therefore,
$$ Au = \alpha u - \beta v $$ $$ Av = \beta u + \alpha v $$
Let $U = \text{span}\{u,v\}$ in $\mathbb{R}^{n}$ be a two dimensional invariant subspace of $A$. Choose coordinate system $(y_1,y_2)$ in $U$ such that every vector $x \in U$ is uniquely represented by
$$ x = y_1 u + y_2 v $$
Suppose $x(t) \in U$ for all $t \in \mathbb{R}$ is a solution to the dynamics $\dot{x}=Ax$ and $AU \subset U$. Show that $y(t)$ satisies an ODE $\dot{y}=By$ and find $B \in \mathbb{R}^{2 \times 2}$.