7.6-16 Boundary cases and defective dynamics

Boundary cases and defective dynamics

Examine the boundary cases where eigenvalue lists alone can mislead a dynamics calculation. Derive Jordan-block powers and compare transient growth, eventual decay and continuous nilpotent evolution.

What you will practise

• Detect polynomial growth at unit modulus
• Account for the initial state
• Separate transient amplification from long-run behavior
• Compare discrete powers with continuous exponentials

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix powers, defective eigenspaces and basic 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 one nilpotent matrix in two different time models and explains why their trajectories behave differently. 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.