A factorization of an m×n matrix A (real or complex) as A = U Σ V*, where U and V are unitary (orthogonal in the real case) matrices and Σ is a diagonal (rectangular) matrix with nonnegative real entries called singular values; it encodes the action of A on orthogonal directions.

Principle

Diagonalize the positive semidefinite matrix A* A to obtain orthonormal right singular vectors and nonnegative singular values (square roots of A* A eigenvalues); left singular vectors are obtained by applying A to right singular vectors and normalizing.

Demonstration

For a 2×2 matrix A, compute A* A, find its eigenpairs (v_i, λ_i), set σ_i = √λ_i, take V whose columns are v_i, form U = A V Σ^−1 on the nonzero singular values, yielding A = U Σ V*; truncating small σ_i gives a best low-rank approximation.

Misapplication

Assuming U, Σ, V are uniquely determined disregards sign and order ambiguities for singular vectors and equal singular values; using SVD blindly for non-linear problems or interpreting small singular values as numerical zeros without error analysis is also misleading.

Consequence

Provides canonical orthonormal bases for domain and codomain aligned to A's action, underpins optimal low-rank approximation (Eckart–Young), stable computation of pseudoinverses, conditioning analysis, and many data-analytic techniques (principal components, latent-factor models).

Reversal

Eigen-decomposition of A itself is the reverse idea but only applicable when A is normal (e.g., symmetric); reversing SVD would discard the orthogonal factorization and with it the direct geometry of input-to-output stretching along orthogonal axes.

Boundary

Exists for every finite matrix over R or C; in infinite-dimensional settings an analogous decomposition requires compact operators and additional functional-analytic hypotheses; SVD is linear-algebraic and does not directly generalize to nonlinear maps.

Semantic Tension

Competes with eigenvalue decomposition when matrix is square and normal; SVD always exists and gives singular directions even when eigen-decomposition fails or is non-orthogonal, but eigen-decomposition conveys spectrum-specific algebraic structure absent from SVD.

Synthesis

A universal orthogonal factorization A = U Σ V* that exposes the magnitudes and directions by which A stretches orthogonal input directions, instrumental for optimal approximation, pseudoinversion, and numerical conditioning.