Complex numbers frequently arise in physical problems,
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 a prime, such as 
, to denote the complex conjugate
of the transposed vector .Now write the positive quadratic form as
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(34) |
After equation (4),
we minimized a quadratic form 
by setting to zero both
and 
.We noted that only one of
and is necessarily zero
because they are conjugates of each other.
Now take the derivative of Q
with respect to the (possibly complex, row) vector .Notice that 
is the complex conjugate transpose
of 
.Thus, setting one to zero sets the other also to zero.
Setting 
gives the normal equations:
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(35) |