Estimation with singular inverses

The classical estimation procedure according to Gauss uses the inversion of non-singular matrices. The author has earlier presented a number of papers concerning the stochastical estimation with the use of singular inverses (Bjerhammar, 1948–1958). The present paper gives a review of the earlier contributions and an application to selected problems. For a geodetic network all points can be considered unknown and an estimate with a generalized inverse gives the minimum variance of the observations as well as the unknowns. This estimate is invariant with respect to rotations and translations, and therefore of special interest for the study of the optimum network. It is furthermore of interest because it eliminates the dependence on the coordinate system. A number of similar applications can be found in most sciences.