Model fitting by least squares

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## Differentiation by a complex vector

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:

 (45)

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:

 (46)

The result is merely the complex form of our earlier result (43). Therefore, differentiating by a complex vector is an abstract concept, but it gives the same set of equations as differentiating by each scalar component, and it saves much clutter.

 Model fitting by least squares

Next: From the frequency domain Up: MULTIVARIATE LEAST SQUARES Previous: Normal equations

2014-12-01