International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 1, Pages 41-46

A novel interpretation of least squares solution

Jack-Kang Chan

Norden Systems, 75 Maxess Road, Melville, New York 11747, USA

Received 5 March 1991; Revised 21 August 1991

Copyright © 1992 Jack-Kang Chan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We show that the well-known least squares (LS) solution of an overdetermined system of linear equations is a convex combination of all the non-trivial solutions weighed by the squares of the corresponding denominator determinants of the Cramer's rule. This Least Squares Decomposition (LSD) gives an alternate statistical interpretation of least squares, as well as another geometric meaning. Furthermore, when the singular values of the matrix of the overdetermined system are not small, the LSD may be able to provide flexible solutions. As an illustration, we apply the LSD to interpret the LS-solution in the problem of source localization.