belongs to a zero eigenvalue of CC T . This therefore completes the proof that matrices A and B satisfy

(M.5).

Proof of Equation (16.39)

We prove here equation (16.39), which states that the Hilbert transform of the complex EOF coef¬cient

± = (X + i XH )† is equal to the coef¬cient itself multiplied by ’i. First note that, if Y = X + i X H ,

then by (16.23) and (16.24)

Y H = (X + i X H )H = x H ’ i X = (’i)(X + i X H ) = ’i Y. (M.18)

By repeatedly using (M.18), we infer that

[Y H ]— = [(’i)Y ]— = i Y — = i/(’i)[Y — ]H = ’[Y — ]H .

Then, with (M.18), we have

H H

± H = Y† p = Y† p = ’[YH ]† p = [i Y]† p = (’i)Y† p = ’i±.

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