Singular Value Decomposition Calculator

Decompose matrices into U, Σ, and V^T components

Enter your matrix to compute its Singular Value Decomposition (SVD). This tool decomposes any m×n matrix A into A = UΣV^T.

Enter matrix elements separated by commas for rows and semicolons for columns (e.g., 1,2,3;4,5,6 for a 2×3 matrix)

SVD Examples

Try these example matrices to understand SVD decomposition

2×2 Identity Matrix

2x2

Simple identity matrix decomposition

Matrix: 1,0;0,1

Size: 2×2

3×2 Rectangular Matrix

3x2

Decomposition of a rectangular matrix

Matrix: 1,2;3,4;5,6

Size: 3×2

2×3 Data Matrix

2x3

Typical data matrix for dimensionality reduction

Matrix: 4,0,3;0,2,0

Size: 2×3

3×3 Diagonal Matrix

diagonal

Diagonal matrix with distinct eigenvalues

Matrix: 3,0,0;0,2,0;0,0,1

Size: 3×3

Other Titles
Understanding Singular Value Decomposition: A Comprehensive Guide
Master the fundamentals of SVD and its applications in data science and linear algebra

What is Singular Value Decomposition (SVD)?

  • Mathematical Foundation
  • Matrix Decomposition Theory
  • Relationship to Eigendecomposition
Singular Value Decomposition (SVD) is a fundamental matrix factorization technique in linear algebra that decomposes any m×n matrix A into three matrices: A = UΣV^T, where U and V are orthogonal matrices and Σ is a diagonal matrix containing singular values.
Mathematical Foundation
For any real matrix A of size m×n, SVD produces: U (m×m orthogonal matrix), Σ (m×n diagonal matrix with non-negative singular values), and V^T (n×n orthogonal matrix transpose). The singular values in Σ are arranged in descending order.
Matrix Decomposition Theory
SVD exists for every matrix, unlike eigendecomposition which only exists for square matrices. The columns of U are called left singular vectors, the columns of V are right singular vectors, and the diagonal elements of Σ are singular values.
Relationship to Eigendecomposition
SVD is closely related to eigendecomposition: the columns of V are eigenvectors of A^TA, columns of U are eigenvectors of AA^T, and singular values are square roots of eigenvalues of A^TA (or AA^T).

Basic SVD Examples

  • For matrix A = [[3,2,2],[2,3,-2]], SVD gives U, Σ, and V^T matrices
  • The largest singular value represents the maximum stretching factor of the linear transformation

Step-by-Step Guide to Computing SVD

  • Manual Calculation Process
  • Numerical Methods
  • Software Implementation
Computing SVD manually involves several steps: calculating A^TA and AA^T, finding their eigenvalues and eigenvectors, constructing V and U matrices, and determining singular values from eigenvalues.
Manual Calculation Process
1. Compute A^TA and find its eigenvalues λ₁, λ₂, ..., λₙ. 2. Calculate singular values σᵢ = √λᵢ. 3. Find eigenvectors of A^TA to form V matrix. 4. Compute U using U = AV/σ for each singular value. 5. Arrange components in descending order of singular values.
Numerical Methods
Modern algorithms use iterative methods like the Golub-Reinsch algorithm or Jacobi methods to compute SVD efficiently. These methods avoid explicitly computing A^TA or AA^T to maintain numerical stability.
Software Implementation
Most mathematical software packages provide built-in SVD functions. Our calculator uses optimized numerical algorithms to compute SVD accurately for matrices up to reasonable sizes while maintaining precision.

Calculation Examples

  • Step-by-step calculation for 2×2 matrix [[1,2],[3,4]]
  • Comparison of different numerical methods for large matrices

Real-World Applications of SVD

  • Data Compression
  • Dimensionality Reduction
  • Recommendation Systems
SVD has numerous practical applications across different fields, from image compression and data analysis to machine learning and signal processing. Its ability to capture the most important features of data makes it invaluable for many computational tasks.
Data Compression
SVD enables lossy data compression by keeping only the largest singular values and their corresponding vectors. This truncated SVD preserves the most significant information while reducing storage requirements significantly.
Dimensionality Reduction
In machine learning, SVD is used for Principal Component Analysis (PCA) to reduce data dimensionality while preserving variance. This helps in visualization, noise reduction, and computational efficiency.
Recommendation Systems
Collaborative filtering systems use SVD to decompose user-item rating matrices, identifying latent factors that explain user preferences and item characteristics for making recommendations.

Application Examples

  • Image compression reducing file size by 90% while maintaining visual quality
  • Netflix recommendation system using SVD for personalized movie suggestions

Common Misconceptions and Correct Methods

  • Uniqueness of SVD
  • Computational Complexity
  • Interpretation of Results
Several misconceptions exist about SVD, particularly regarding its uniqueness, computational requirements, and result interpretation. Understanding these clarifies when and how to use SVD effectively.
Uniqueness of SVD
While singular values are unique (up to ordering), the U and V matrices are not unique when singular values are repeated. The span of columns corresponding to equal singular values is uniquely determined, but individual vectors within that space can vary.
Computational Complexity
SVD computation is O(min(m²n, mn²)) for an m×n matrix, not O(n³) as sometimes believed. For large matrices, randomized SVD algorithms can provide good approximations much faster.
Interpretation of Results
Singular values indicate the 'importance' of each component, but their absolute magnitudes depend on data scaling. Always consider the ratio of singular values rather than absolute values when assessing component significance.

Clarification Examples

  • Different U,V matrices for the same matrix with repeated singular values
  • Scaling effects on singular value magnitudes

Mathematical Derivation and Advanced Examples

  • Theoretical Derivation
  • Geometric Interpretation
  • Advanced Applications
The mathematical foundation of SVD stems from the spectral theorem and optimization theory. Understanding the derivation provides deeper insight into why SVD works and how to interpret its results.
Theoretical Derivation
SVD can be derived from the variational characterization: σ₁ = max ||Ax||/||x|| over unit vectors x. This maximum is achieved by the first right singular vector, and subsequent singular values are found by imposing orthogonality constraints.
Geometric Interpretation
Geometrically, SVD represents any linear transformation as a composition of rotation (V^T), scaling (Σ), and another rotation (U). This decomposition reveals how the transformation affects different directions in space.
Advanced Applications
Advanced applications include solving least squares problems, computing matrix pseudoinverses, analyzing network structures, and solving partial differential equations using proper orthogonal decomposition (POD).

Advanced Examples

  • Geometric visualization of 2D linear transformation via SVD
  • Using SVD pseudoinverse for solving overdetermined systems