Σ SVD Calculator

SVD — Singular Value Decomposition

Singular value decomposition (SVD) factors any matrix A into UΣVᵀ where U and V are orthogonal and Σ contains the singular values, and this calculator computes all three matrices. Shows U (left singular vectors), Σ (singular values), and V (right singular vectors) with rank and condition number.

Σ

Singular Value Decomposition A = UΣVᵀ

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Matrix A
Quick Examples

Singular Value Decomposition

SVD is one of the most important matrix factorizations. Every real m×n matrix A can be written as:

$$A = U\Sigma V^T$$
  • U: m×m orthogonal matrix (left singular vectors)
  • Σ: m×n diagonal matrix (non-negative singular values σ₁ ≥ σ₂ ≥ ... ≥ 0)
  • V: n×n orthogonal matrix (right singular vectors)

Applications

  • Image compression (low-rank approximation)
  • Principal Component Analysis (PCA)
  • Pseudoinverse and least-squares solutions
  • Numerical rank determination
  • Condition number: κ = σ₁/σₙ (ratio of largest to smallest non-zero singular value)

What Do U, Σ, and V Mean?

U and V are orthogonal matrices (their columns are perpendicular unit vectors), and Σ (sigma) is a diagonal matrix containing the "singular values" — numbers that describe how much the matrix stretches space in each direction. Unlike eigenvalues, SVD works on any matrix, not just square ones.

Worked Example (2×2)

For a diagonal matrix like A = [[3,0],[0,2]], SVD is straightforward: U and V are both the identity matrix, and Σ = [[3,0],[0,2]] — the singular values are simply the diagonal entries, since the matrix doesn't rotate anything, only stretches it.

Frequently Asked Questions

Eigenvalue decomposition only works on square matrices and doesn't always exist. SVD works on every matrix, of any shape, which makes it more broadly useful in practice.
Singular values indicate how important each "direction" in the data is. The largest singular values capture the most significant patterns, which is the basis for compressing images and reducing large datasets.
Yes — SVD is one of the few decompositions that works on any m×n matrix, regardless of whether it's square, which is one reason it's so widely used.

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