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André-Louis Cholesky was a mathematician and military surveyor whose computational work grew out of geodesy. During the revision of French triangulation surveys, he developed the method now known as the Cholesky decomposition.^[ref] For a real symmetric positive-definite matrix $A$, the Cholesky decomposition writes $A=L{L}^{\mathsf{T}}$ with $L$ lower triangular matrix and positive diagonal entries. A linear system can then be solved by two triangular substitutions. The link with surveying is natural: reconciling redundant, imperfect measurements by least squares leads to such systems when the design matrix has full column rank.

Identity source: www.polytechnique.edu/bibliotheque/actualites/cholesky-andre-x1895

André-Louis Cholesky developed his calculation in the setting of geodetic adjustment: measured angles and lengths must satisfy geometric relations, while corrections should be small. The manuscript reproduced in Claude Brezinski's study is titled Sur la résolution numérique des systèmes d’équations linéaires and ends with the date 2 December 1910 at Vincennes. The manuscript reports a 56-equation calculation adjusting altitudes in an Algerian triangulation network. Its date establishes when this written account existed, rather than an exact invention date.^[ref]

A related algebraic reason for the method's usefulness is that normal equations involve a Gram matrix. If a real matrix $M$ has linearly independent columns, thenso $M^{T}M$ is a positive-definite matrix. That gives the structure needed by a Cholesky decomposition. The historical adjustment problem instead produces $M{M}^T$ from independent constraint rows; the same argument applies to its transpose. In numerical least squares, explicitly forming normal equations squares the condition number of a full-column-rank design matrix; a QR decomposition can therefore be preferable for accuracy. Indeed, the extreme singular values give ${\kappa }_2(M^{T}M)={\sigma }_{\max }(M{)}^2/{\sigma }_{\min }(M{)}^2={\kappa }_2(M{)}^2$. The factorization remains useful when a positive-definite matrix is already the natural input.

Historical research article by Claude Brezinski, published in Revue d’histoire des mathématiques 11 (2005), pp. 205–238. Numdam hosts the original journal article. The printed received date, 4 November 2004, and revised date, 11 May 2005, are distinct from publication.

Original article: www.numdam.org/item/RHM_2005__11_2_205_0.pdf

Section 4 reproduces a primary manuscript authored by André-Louis Cholesky and dated Vincennes, 2 December 1910. Cholesky is the author of that manuscript; Brezinski is the author of the surrounding 2005 article. The manuscript date does not establish an exact invention date or a broad priority claim.

Cited passages: pp. 210–211: constrained geodetic corrections and the matrix $M{M}^T$; p. 217: manuscript title, 2 December 1910 date and uncertainty about the exact invention date; §4, p. 229: the primary manuscript’s reported 56-equation altitude adjustment for Algerian triangulation and closing date. The primary manuscript transcription was inspected as text; no inspection of handwriting images is claimed.