Overview
Eigenvalue Decomposition finds eigenvectors and eigenvalues that reveal principal directions, modes, or repeated matrix behavior.
ExtendedSource checked
Eigenvalue Decomposition
Represents a matrix through directions that are only scaled by the matrix.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Advanced
- Family
- Matrix Decomposition
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 6
- Code example
- Ready
- Visualization
- Ready
Core Concepts
eigenvectoreigenvaluespectral basismatrix mode
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n^3)
Average
O(n^3)
Worst
O(n^3)
Space
O(n^2)
When to Use
- Use when matrix modes or principal directions are the question.
- Use when principal component analysis is a central requirement.
When Not to Use
- Avoid when only a linear solve is needed and factorization is cheaper.
- Avoid when roundoff, conditioning, or discontinuities dominate the result.
What It Does
- Finds eigenvectors
- Finds eigenvalues
- Represents a matrix in a modal basis
Use Cases
- Principal component analysis
- Stability analysis
- Spectral graph methods
Inputs
- Square matrix
Outputs
- Eigenvalues
- Eigenvectors
Key Formulas (1)
Eigen equation
A · v = λ · v
A * v = lambda * vDefines a vector whose direction is preserved by the matrix.
Pseudocode
- 1.Choose a suitable numerical method
- 2.Iteratively reduce or transform the matrix
- 3.Extract eigenvalues
- 4.Compute corresponding eigenvectors
- 5.Validate residuals
Canonical Example
- Input
- matrix scales x by 3 and y by 2
- Output
- eigenvalues = 3 and 2
- Explanation
- The coordinate axes are eigenvectors because they keep their directions.
Implementation Notes
- Check residual norm A*v - lambda*v for computed pairs.
- Compare against a small hand-checkable case before scaling up.
- Keep a tiny example with expected output beside the implementation.
Common Pitfalls
- Repeated or nearly equal eigenvalues can make eigenvectors unstable.
- Ignoring conditioning can make a correct method return unreliable numbers.
- Not defining boundary, empty, or degenerate inputs before implementation.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow vectors that keep direction while being scaled by a matrix transform.
Step 1 of 5: Choose a suitable numerical method
numeric-iterationNotes
General nonsymmetric matrices can have complex eigenvalues and less stable eigenvectors.
Sources (1)
What each reference supports for this entry.
Matrix Computations
BookNumerical linear algebra reference covering eigenvalue decomposition and matrix algorithms.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Often paired withQR Decomposition
Numerical Methods · Core
QR iteration is a classic path to eigenvalue computation.
Compare with thisGeneralizesPower Iteration
Numerical Methods · Core
Power iteration estimates a dominant eigenpair.
Compare with thisRelationships
View full relationship pageGeneralizesPower Iteration
Power iteration estimates a dominant eigenpair.
Conceptual neighborSingular Value Decomposition
Both expose important matrix directions and scales.
Related Algorithms
Tags
Numerical MethodsLinear AlgebraDecomposition