7.6-11 Continuous-time stability

Continuous-time stability

Classify continuous-time linear stability using real parts of eigenvalues. Separate asymptotic decay, instability and bounded nondecaying motion, and pay special attention to defective boundary cases.

What you will practise

• Apply the real-part criterion
• Identify sinks and saddles
• Treat imaginary-axis modes
• Analyze nilpotent and defective examples

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review continuous eigenmodes and basic matrix exponentials. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently.

The final comparison uses identical eigenvalue lists with different Jordan behavior to show why boundary stability requires more information. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.