7.6-04 Discrete growth and decay

Discrete growth and decay

Predict long-run discrete behavior from eigenvalue moduli under explicit diagonalizability assumptions. Distinguish decay, growth, bounded oscillation and convergence, while accounting for the initial state.

What you will practise

• Use modulus rather than sign alone
• Identify stable eigenspaces
• Separate boundedness from convergence
• Check complex eigenvalue magnitudes

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, eigenmodes and complex modulus. 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 last case challenges reliance on the real part of a complex eigenvalue when the model advances in discrete steps. 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.