Overview
Singular Value Decomposition supports dimensionality reduction, pseudoinverses, low-rank approximation, and stability analysis.
CoreSource checked
Singular Value Decomposition
Factors a matrix into singular vectors and singular values that reveal its main directions.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Advanced
- Family
- Matrix Decomposition
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 3
- Code example
- Ready
- Visualization
- Ready
- Aliases
- SVD
Core Concepts
singular vectorsingular valuelow-rank approximationorthogonal factor
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 solving equations needs singular vector.
- Use when the input matches the assumptions behind matrix Decomposition.
When Not to Use
- Avoid when the problem is ill-conditioned and no scaling or stability checks are available.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Factors a matrix into singular vectors and singular values that reveal its main directions
- Uses singular vector and singular value to guide the work.
- Produces approximate numeric result.
Use Cases
- Solving equations
- Approximating integrals
- Working with matrices and differential equations
Inputs
- Numeric function, matrix, or sampled data
- Initial estimate or discretization
- Tolerance
Outputs
- Approximate numeric result
- Error estimate or residual
Key Formulas (1)
SVD factorization
A = U Σ VT
A = U Sigma V^TDecomposes a matrix into singular vectors and singular values.
Pseudocode
- 1.Prepare the inputs and state needed for Singular Value Decomposition.
- 2.Apply the singular vector step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return approximate numeric result.
Canonical Example
- Input
- Singular Value Decomposition input with a small representative case
- Output
- Approximate numeric result
- Explanation
- Singular Value Decomposition applies singular vector to produce approximate numeric result.
Implementation Notes
- Check convergence with residuals or estimated error, not iteration count alone.
- Keep the singular vector step separate enough to test independently.
Common Pitfalls
- Ignoring conditioning and residuals can produce plausible-looking but wrong numeric answers.
- Forgetting to check edge cases can make Singular Value Decomposition look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow the numeric approximation improving against the target function, matrix, or state.
Step 1 of 4: Prepare the inputs and state needed for Singular Value Decomposition.
numerical-methods-mechanismNotes
SVD is robust and informative but usually more expensive than simpler factorizations.
Sources (1)
What each reference supports for this entry.
Matrix Computations
BookReference text for numerical linear algebra, matrix factorizations, SVD, and eigenvalue methods.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageConceptual neighborEigenvalue Decomposition
SVD generalizes spectral analysis to rectangular matrices.
Used byPrincipal Component Analysis
PCA is commonly computed through SVD.
Related Algorithms
Tags
Numerical MethodsLinear AlgebraDecompositionDimensionality Reduction