SVD Calculator (Singular Value Decomposition)

Singular value decomposition is a factorization of a real or complex matrix A of size m×n into the product A = UΣVᵀ, where U is an m×m orthogonal matrix, Σ is an m×n diagonal matrix of non-negative values called singular values, and V is an n×n orthogonal matrix. It generalizes the concept of eigendecomposition to non-square matrices.

No, the SVD is not unique in general. The singular values themselves are unique and always listed in non-increasing order (σ₁ ≥ σ₂ ≥ … ≥ 0), but the columns of U and V (the singular vectors) can differ by sign flips or, when singular values are repeated, by orthogonal rotations in the corresponding subspace.

SVD decomposes any linear transformation (matrix) into three geometric steps: a rotation/reflection (Vᵀ), a scaling along principal axes (Σ), and another rotation/reflection (U). This reveals the fundamental structure of the transformation — how much it stretches space in each independent direction, which is captured by the singular values.

For a symmetric positive definite matrix A, the SVD coincides with its eigendecomposition: A = QΛQᵀ. The singular values equal the eigenvalues, and the left and right singular vectors are the same orthonormal eigenvectors. This makes symmetric matrices a special, elegant case of SVD.

If A is a unitary (or orthogonal) matrix, all its singular values are exactly 1. This is because a unitary matrix preserves vector lengths, so the scaling matrix Σ is simply the identity. The matrices U and V are also orthogonal, so A = UIVᵀ.

The rank of matrix A equals the number of non-zero singular values in Σ. Numerically, a singular value is treated as zero if it falls below a small tolerance threshold (e.g., 1e-9). This makes SVD the most reliable numerical method for determining matrix rank.

The condition number is the ratio of the largest singular value to the smallest non-zero singular value (σ₁ / σₘᵢₙ). A large condition number indicates a nearly singular or ill-conditioned matrix, meaning small changes in input can cause large changes in the output — a critical property for numerical stability in linear solvers.

SVD is used across many fields: in data science and machine learning for Principal Component Analysis (PCA) and dimensionality reduction; in image compression to approximate images with fewer singular values; in natural language processing for Latent Semantic Analysis (LSA); and in numerical linear algebra for computing pseudoinverses and solving least-squares problems.