A factorization of a finite-dimensional linear map (or matrix) A as A = U Σ V^*, where U and V are unitary (or orthogonal) matrices and Σ is a diagonal matrix with nonnegative real entries called singular values.

Principle

It diagonalizes the positive semidefinite operators A^*A and AA^*, producing orthonormal bases of domain and codomain that optimally diagonalize A in the least-squares sense and always exists for any rectangular matrix.

Demonstration

For a 2×2 real matrix A one computes eigenvectors of A^T A to obtain V, sets singular values as square roots of eigenvalues, and obtains U from U = A V Σ^{-1} for nonzero singular values; the largest singular value equals the operator norm of A.

Misapplication

Interpreting singular vectors as eigenvectors of A itself (rather than of A^*A or AA^*) or using Σ's ordering arbitrarily can mislead about rotations vs scalings and give incorrect low-rank approximations.

Consequence

SVD yields the matrix rank, spectral norms, stable numerical pseudoinverses, and best low-rank approximations (Eckart–Young property) used in data compression, denoising and principal subspace extraction.

Reversal

An eigenvalue decomposition applies to a diagonalizable square matrix with respect to a single basis; SVD is a two-basis factorization applicable even when eigen decomposition fails or the matrix is rectangular.

Boundary

Applies to linear maps between finite-dimensional inner-product spaces (or compact operators in separable Hilbert spaces); in infinite dimensions existence requires compactness and singular values accumulate to zero.

Semantic Tension

Often contrasted with eigenvalue decomposition: eigenpairs describe action on one space relative to a single basis and may be complex, while SVD gives orthonormal input/output bases and nonnegative singular values for any real rectangular matrix.

Synthesis

SVD expresses any finite linear map as a composition of an orthogonal/unitary rotation, an axis-aligned nonnegative scaling by singular values, and another rotation, providing robust spectral information and optimal approximations.