Singular Value Decomposition (SVD) Calculator

The SVD factorizes any real m by n matrix A into a product A = U Sigma V transpose. Sigma is a diagonal matrix of non-negative singular values sigma_1 >= sigma_2 >= ... that measure how much A stretches space along each principal direction. U holds the left singular vectors and V holds the right singular vectors, both with orthonormal columns. Unlike eigendecomposition, the SVD exists for every matrix, square or not.

It uses the classic eigendecomposition route. First it forms the symmetric matrix A transpose times A, then diagonalizes it with the Jacobi eigenvalue algorithm. The singular values are the square roots of those eigenvalues, the right singular vectors are the eigenvectors, and each left singular vector is u = A v / sigma. This returns the same factorization a library like NumPy gives, up to the sign of paired vectors.

The right singular vectors v_i live in the input space, the columns of V and the eigenvectors of A transpose times A. The left singular vectors u_i live in the output space, the columns of U. They are linked by A v_i = sigma_i u_i, so A maps each right singular vector onto its left partner, scaled by the singular value. Both sets are orthonormal, which is what makes the SVD so useful for rotations, projections, and least squares.

The rank equals the number of non-zero singular values. The calculator counts every singular value above a small tolerance that scales with the largest singular value and the matrix size, so floating-point noise near zero is not mistaken for a real direction. If some singular values are effectively zero, the matrix is rank deficient and collapses those directions to nothing.

The 2-norm condition number is the ratio of the largest singular value to the smallest, sigma_max divided by sigma_min. It measures how much the matrix can amplify errors when you solve a linear system: a value near 1 is well conditioned, while a very large value warns that the answer is sensitive to small input changes. When the smallest singular value is zero, the matrix is singular and the condition number is infinite, which the calculator reports instead of dividing by zero.

Forming A transpose times A turns the problem into a symmetric eigenproblem, which the Jacobi method solves cleanly with guaranteed convergence and orthonormal vectors. The trade-off is precision: squaring the matrix also squares its condition number, so a true-zero singular value surfaces near the square root of machine epsilon rather than exactly zero. For the integer and small decimal matrices this calculator handles, that floor sits far below any real singular value, so the rank and singular values stay accurate.