### Dynamics with complex Eigenvalues

Due by Mon, 16 Apr 2018

Please do not submit this assignment. This is to only help with understanding the material on ODEs. It is not a model for final examination questions.

Let $A : \mathbb{R}^2 \to \mathbb{R}^2$ be a linear transformation. We can also regard this as a linear transformation $A : \mathbb{C}^2 \to \mathbb{C}^2$.

• For what values of $tr(A)$ and $det(A)$ does $A$ have eigenvalues $\alpha \pm i\beta$ for $\alpha, \beta \in\mathbb{R}$ with $\beta\neq 0$?

Assume henceforth that $A$ has complex eigenvalues $\alpha \pm i\beta$ for $\alpha, \beta \in\mathbb{R}$ with $\beta\neq 0$.

• Let $\lambda = \alpha + i\beta$ and let $v\in\mathbb{C}^2$ be an eigenvector of $A$ with $Av = \lambda v$. Show that its (component-wise) complex conjugate $\bar{v}$ is also an eignevector.
• Let $w_1 = (v + \bar{v})/2$ and $w_2 = (v - \bar{v})/2i$. Show that $w_1, w_2 \in\mathbb{R}^2\subset{\mathbb{C}}^2$ and $w_1$, $w_2$ form a basis of $\mathbb{R}^2$ (as a real vector space).
• Find the matrix of $A$ with respect to the basis $w_1$, $w_2$.
• Recall that $\mathbb{C}$ is a real vector space with basis $1, i$. Let $P: \mathbb{R}^2 \to \mathbb{C}$ be the (invertible) linear transformation such that $P(w_1) = 1$ and $P(w_2) = i$. Let $B = P\circ A \circ P^{-1}$. Show that $B(z) = \lambda z$ for all $z \in \mathbb{C}$.