NOTES ON STABILITY ANALYSIS LANCE D. DRAGER 1. Introduction These notes are a direct continuation of the notes “Notes on Linear Systems of Differential Equations.” In these notes, we will study the solutions of a linear system of differential equations with constant coefficients, dy dt = Ay. We want to study the long time behavior of the solutions, in other words, what happens to y(t) as t →∞. We begin with some preliminary material. 1.1. Norms on Vector Spaces. We want to measure the “size” or “length” of a vector. The mathematical device for doing this in called a norm. Here is the definition. Definition 1.1. Let V be a vector space over K. 1 A norm on V is a function ‖·‖ : V → R : v →‖v‖ that has the following properties. (1) ‖v‖≥ 0 for all v ∈ V , and ‖v‖ = 0 if and only if v = 0. (2) If λ ∈ K and v ∈ V , ‖λv‖ = |λ|‖v‖. (3) For all v 1 ,v 2 ∈ V , ‖v 1 + v 2 ‖≤‖v 1 ‖ + ‖v 2 ‖. This is called the triangle inequality. Exercise 1.2. Show that ‖−v‖ = ‖v‖ and ‖v 1 − v 2 ‖ = ‖v 2 − v 1 ‖. The following is a simple consequence of the triangle inequality, which we will include under that name. Proposition 1.3. For all v 1 ,v 2 ∈ V ,   ‖v 1 ‖−‖v 2 ‖   ≤‖v 1 ± v 2 ‖. Proof. We have ‖v 1 ‖ = ‖v 1 − v 2 + v 2 ‖ ≤‖v 1 − v 2 ‖ + ‖v 2 ‖, by the triangle inequality, so ‖v 1 ‖−‖v 2 ‖≤‖v 1 − v 2 ‖. Switching the roles of v 1 and v 2 gives ‖v 2 ‖−‖v 1 ‖≤‖v 2 − v 1 ‖ = ‖v 1 − v 2 ‖. Thus, we have ±  ‖v 1 ‖−‖v 2 ‖  ≤‖v 1 − v 2 ‖, 1 As usual, K stands for either R or C. 1