Model fitting by least squares |

Complex numbers frequently arise in physical applications,
particularly those with Fourier series.
Let us extend the multivariable least-squares theory
to the use of complex-valued unknowns .
First,
recall how complex numbers were handled
with single-variable least squares;
i.e., as in the discussion leading up to equation (5).
Use an asterisk, such as , to denote the complex conjugate
of the transposed vector .
Now, write the positive **quadratic form** as:

Recall from equation (4), where
we minimized a quadratic form
by setting to zero, both
and
.
We noted that only one of
and
is necessarily zero,
because these terms are conjugates of each other.
Now, take the derivative of
with respect to the (possibly complex, row) vector .
Notice that
is the complex conjugate transpose
of
.
Thus, setting one to zero also sets the other to zero.
Setting
gives the normal equations:

Model fitting by least squares |

2014-12-01