Power Iteration
Numerical method to find the dominant eigenvector.
Key idea
Numerical method to find the dominant eigenvector.
In depth
Repeatedly multiply a vector by A and normalise: it converges to the eigenvector of the dominant eigenvalue (if unique). It is simple and used in PageRank for huge sparse matrices.
Worked example
A = [2 0; 0 1], v = (1, 1): after k steps v ∝ (2ᵏ, 1) → direction (1, 0).
Real-world uses
- Finding the most important direction in a large network or a PageRank-style ranking quickly
Common mistake
Starting with a vector orthogonal to the dominant eigenvector.
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 Power Iteration?
Numerical method to find the dominant eigenvector. It is a topic at the expert level in Linear Algebra.
Where is Power Iteration used in real life?
Real-world examples: Finding the most important direction in a large network or a PageRank-style ranking quickly.
What should I learn before Power Iteration?
Make sure you are comfortable with Condition Number, the previous topic in Linear Algebra.
What is a common mistake with Power Iteration?
Starting with a vector orthogonal to the dominant eigenvector.
How can I practise Power Iteration?
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 Power Iteration?
Move on to Gershgorin Circle Theorem 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.