Who really invented the SVD? Tracing linear algebra's workhorse to the 1870s

G.W. Stewart’s 1993 essay reconstructs how the singular value decomposition — now a staple of numerical computing, statistics, and machine learning — was discovered and rediscovered by several mathematicians working independently across four decades. The story begins not with matrices but with the theory of bilinear and quadratic forms. Eugenio Beltrami (1873) and Camille Jordan (1874) each derived, by different routes, the result that a form can be reduced to diagonal shape by orthogonal substitutions — the SVD in disguise. James Joseph Sylvester arrived at essentially the same theorem again in the late 1880s, apparently unaware of the earlier work, reflecting how fragmented mathematical communication was before modern citation practice.

The decomposition took on deeper significance in the early 20th century. Erhard Schmidt (1907) extended the idea to integral equations and infinite-dimensional settings, and in doing so established the approximation property that makes the SVD so useful today: the best low-rank approximation to an operator is obtained by truncating its singular values — a result later rediscovered in matrix form by Eckart and Young. Hermann Weyl (1912) then developed a general perturbation theory, connecting singular values to min-max characterizations and bounding how they shift under perturbation. Stewart also traces the tangled history of terminology and the link to the eigenvalues of A-transpose-times-A.

The broader point is a lesson in how mathematical ideas actually propagate: a single concept surfaced repeatedly under different names and motivations, and only accreted its modern meaning and computational importance over time. Note for editorial context — this is a history-of-mathematics paper with no cybersecurity angle; it surfaced on Hacker News on its technical and historical merits, and is relevant to this audience mainly because the SVD underpins modern data analysis, dimensionality reduction, and AI.