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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