Singular Values Calculator - SVD Matrix Decomposition

Singular Value Decomposition (SVD) is one of the most important factorizations in linear algebra. For any real m×n matrix A, SVD expresses A as the product A = UΣV^T, where U is an m×m orthogonal matrix, Σ is an m×n diagonal matrix with non-negative entries on the diagonal, and V is an n×n orthogonal matrix. The diagonal entries σ₁ ≥ σ₂ ≥ … ≥ 0 of Σ are the singular values of A. Singular values are always non-negative real numbers, even when the original matrix contains negative entries or complex structure. They are uniquely determined by the matrix — unlike the vectors U and V, which may not be unique when singular values repeat. The number of non-zero singular values equals the rank of the matrix, which measures the dimension of the column space (the set of all possible outputs of the linear transformation represented by A). The relationship between SVD and eigenvalues is precise: the singular values of A are the square roots of the eigenvalues of the symmetric positive semidefinite matrix A^T·A (or equivalently A·A^T for the left singular vectors). This calculator computes A^T·A and applies the Jacobi eigenvalue algorithm — an iterative method that zeros out off-diagonal elements of a symmetric matrix through a sequence of orthogonal rotations called Givens rotations — to find its eigenvalues, then takes their square roots. The largest singular value σ₁ equals the spectral norm (also called the 2-norm or operator norm) of the matrix — the maximum factor by which the matrix can stretch a unit vector. The Frobenius norm is the square root of the sum of all squared singular values, and it equals the square root of the sum of all squared matrix entries. These norms have practical significance: the Frobenius norm is the most natural measure of total energy in a matrix, while the spectral norm controls sensitivity to perturbations. The condition number κ = σ₁/σₙ (ratio of largest to smallest singular value) quantifies how well-conditioned the matrix is for solving linear systems. A condition number near 1 means the system is well-conditioned and easy to solve accurately. A very large condition number — say above 10⁶ or 10¹² — signals an ill-conditioned system where small perturbations in the input can cause large changes in the solution, a common source of numerical instability in scientific computing and machine learning. SVD has a remarkable range of applications. In data science, Principal Component Analysis (PCA) — the workhorse of dimensionality reduction — is mathematically equivalent to computing the SVD of the centred data matrix. The principal components are the right singular vectors V, and the variance explained by each component is proportional to the square of the corresponding singular value. In image processing and compression, retaining only the k largest singular values and their associated vectors produces a rank-k approximation of the image matrix that minimises the Frobenius norm error among all rank-k matrices — this is the Eckart-Young theorem. In recommendation systems (collaborative filtering), matrix factorisation via SVD is used to predict user ratings for unseen items. In engineering, the pseudo-inverse A† = VΣ†U^T provides the minimum-norm least-squares solution to overdetermined or underdetermined systems, which is critical in robotics, control theory, and signal processing. For this calculator, matrices up to approximately 10×10 are handled reliably by the Jacobi algorithm. Very large matrices or matrices with many near-duplicate singular values may lose some precision due to accumulated floating-point rounding, but results are accurate to at least six significant digits for typical inputs.