1A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation Ax = b, where A is an m×n full-rank matrix, b is a column-vector of dimension m, and m (the number of equations) is larger than or equal to n (the dimension of the unknown vector x). Generally, the equations are inconsistent and there is no feasible solution for x unless b belongs to the column-span of A. In the least-square approach, a candidate solution is found as the unique x that minimizes the error function ‖Ax − b‖2. We propose a more general approach that consist in considering all the consistent subset of the equations, finding their solutions, and taking a weighted average of them to build a candidate solution. In particular, we show that by weighting the solutions with the squared determinant of their coefficient matrix, the resulting candidate solution coincides with the least square solution