Singulärvärdesdekompositionen

Singular values are nonnegative real numbers associated with a real or complex matrix. For an m×n matrix A, the singular values are the square roots of the eigenvalues of A* A (or A^T A in the real case). The singular value decomposition expresses A as A = U Σ V*, where U and V are unitary (orthogonal) matrices and Σ is a diagonal matrix with nonincreasing diagonal entries σ1 ≥ σ2 ≥ … ≥ σr ≥ 0, where r is the rank of A. The σi are the singular values of A. In Swedish, the term for singular value is singulärvärde and the decomposition is singulärvärdesdekomposition.

Properties of the singular values include that the nonzero σi are invariant under multiplication of A by unitary matrices on the left or right. The largest singular value σ1 equals the operator (spectral) norm of A, and the number of positive singular values equals the rank of A. The condition number κ(A) is defined as σ1/σr (or σ1/σk for the smallest nonzero σk). Singular values provide a stable measure of the input-output scaling of A.

Computation and variants: The singular values are computed via the singular value decomposition (SVD). Numerical algorithms typically use Golub–Kahan bidiagonalization to produce U, Σ, and V efficiently and stably.

Applications: SVD enables low-rank approximations A ≈ U_k Σ_k V_k*, with k < min(m,n). This underpins principal component analysis, data compression, noise reduction, and pseudoinverse construction. In solving least-squares problems, the SVD yields the most stable solution and reveals conditioning; the Moore–Penrose pseudoinverse A^+ is V Σ^+ U*, with Σ^+ formed by reciprocals of the nonzero singular values.

Geometric interpretation: Each singular value scales vectors in the right singular subspace along corresponding left singular directions, quantifying the maximal stretching A can perform in orthogonal input directions.