Singulararvot

Singulararvot, or singular values, are nonnegative real numbers that quantify how a linear transformation stretches or contracts space along specific directions. For an m-by-n matrix A, the singular values appear as the diagonal entries σ1 ≥ σ2 ≥ … ≥ σr ≥ 0 of the diagonal matrix Σ in the singular value decomposition A = U Σ V*, where U and V have orthonormal columns (V* denotes the transpose or conjugate transpose). Equivalently, the singular values are the square roots of the eigenvalues of AᵀA (or AAᵀ). The number r is the rank of A and equals the number of positive singular values.

Properties and interpretation: The singular values are invariant under multiplication by orthogonal matrices in the sense that the multiset of singular values of Q A P is the same as that of A for any orthogonal Q and P. The Frobenius norm satisfies ||A||F² = σ1² + σ2² + … + σr², and for a square A, the absolute determinant satisfies |det(A)| = σ1 σ2 … σn. The largest singular value σ1 indicates the maximal stretch, while the ratio σ1/σr defines the condition number, a measure of numerical stability for solving linear systems involving A. Each singular value σi has associated right singular vector vi and left singular vector ui, with A vi = σi ui.

Computation and applications: Singular values are computed via the singular value decomposition, commonly using stable algorithms such as Golub–Kahan bidiagonalization. They are central in applications including dimensionality reduction (PCA-related insights), data compression (reducing redundancy with low-rank approximations A ≈ U Σ V*), image and signal processing, and recommender systems.

History: The concept originated with the development of the singular value decomposition, popularized in linear algebra by Eckart and Young in the 1930s and refined through numerical methods in later decades.