= Cholesky decomposition and geodetic adjustment
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<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.<La méthode de Cholesky by Claude Brezinski>[Claude Brezinski, La méthode de Cholesky, printed pp. 210–211: geodetic least-squares corrections; p. 217: manuscript title and caution about invention date; §4, p. 229: reproduced manuscript's 56-equation adjustment and closing 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, then
$$
v^TM^TMv=\|Mv\|_2^2>0\qquad(v\ne0),
$$
so $M^TM$ is a <positive-definite matrix>. That gives the structure needed by a <Cholesky decomposition>. The historical adjustment problem instead produces $MM^T$ 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 $\kappa_2(M^TM)=\sigma_{\max}(M)^2/\sigma_{\min}(M)^2=\kappa_2(M)^2$. The factorization remains useful when a <positive-definite matrix> is already the natural input.
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