7.6-08 Markov convergence and Perron–Frobenius

Markov convergence and Perron–Frobenius

Understand the conditions behind finite Markov-chain convergence. Use regularity, irreducibility and aperiodicity carefully, and examine counterexamples where a stationary distribution exists without convergence from every start.

What you will practise

• Recognize positive matrix powers
• Distinguish uniqueness from attraction
• Diagnose periodic behavior
• Use self-loops and spectral information

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review stationary distributions, matrix powers and eigenvalue 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 final computation verifies regularity directly through a positive matrix square rather than through an unsupported shortcut. 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.