Differential Equations: Problem Set 11

Let $A$ be a $2 \times 2$ matrix (of constants) and let $λ_{1}, λ_{2}$ be eigenvalues of $A$ . Show that:
1. $λ_{1} + λ_{2} = trace (A)$
2. $λ_{1} \cdot λ_{2} = det (A)$

Consider the first order linear system $x^{'} = A x$ where $A$ is a $2 \times 2$ matrix and $λ_{1}, λ_{2}$ are eigenvalues of $A$ . Recall that $x = 0 = (0, 0)^{T}$ is always an equilibrium of this system. Complete the following table that summarizes the type and stability of this equilibrium in terms of the signs of the trace and determinant of $A$ .

Case	type	stability	$(tr A)^{2} - 4 det A$
$λ_{1} < λ_{2} < 0$	sink	stable	+
$0 < λ_{1} < λ_{2}$
$λ_{1} < 0 < λ_{2}$
$λ_{1, 2} = α \pm i β, α = 0$
$λ_{1, 2} = α \pm i β, α < 0$
$λ_{1, 2} = α \pm i β, α > 0$

Using the information from the table:
1. Make a plot of $det (A)$ vs. $trace (A)$ , labeling each region according to the type and stability of $(0, 0)^{T}$ .
2. Finish the statement of this theorem: $(0, 0)^{T}$ is (asymptotically) stable if and only if ________ and __________ .
(5.1) 2, 4
(5.2) 1-12