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## Normal equations

An important concept is that when energy is minimum, the residual is orthogonal to the fitting functions. The fitting functions are the column vectors , , and .Let us verify only that the dot product vanishes; to do this, we'll show that those two vectors are orthogonal. Energy minimum is found by
 (31)
(To compute the derivative refer to equation (13).) Equation (31) shows that the residual is orthogonal to a fitting function. The fitting functions are the column vectors in the fitting matrix.

The basic least-squares equations are often called the normal" equations. The word normal" means perpendicular. We can rewrite equation (28) to emphasize the perpendicularity. Bring both terms to the left, and recall the definition of the residual from equation (13):
 (32) (33)
Equation (33) says that the residual vector is perpendicular to each row in the matrix. These rows are the fitting functions. Therefore, the residual, after it has been minimized, is perpendicular to all the fitting functions.

Next: Differentiation by a complex Up: MULTIVARIATE LEAST SQUARES Previous: Inside an abstract vector
Stanford Exploration Project
4/27/2004