Markov Chains and Matrices
Transition matrices and steady states.
Key idea
Transition matrices and steady states.
In depth
A transition matrix P (columns sum to 1) moves a probability vector one step. After many steps it often approaches a steady state, an eigenvector for eigenvalue 1.
Worked example
P = [0.9 0.5; 0.1 0.5]: steady state (5/6, 1/6).
Real-world uses
- Modelling customer switching between brands, weather patterns and queueing systems
Common mistake
Mixing up rows and columns when normalising.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Markov Chains and Matrices?
Transition matrices and steady states. It is a topic at the expert level in Linear Algebra.
Where is Markov Chains and Matrices used in real life?
Real-world examples: Modelling customer switching between brands, weather patterns and queueing systems.
What should I learn before Markov Chains and Matrices?
Make sure you are comfortable with Cholesky Decomposition, the previous topic in Linear Algebra.
What is a common mistake with Markov Chains and Matrices?
Mixing up rows and columns when normalising.
How can I practise Markov Chains and Matrices?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Markov Chains and Matrices?
Move on to Google PageRank next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.