Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 138

and complex phase the same as that of the n th component of v , and with remaining
components having null values. In array form,

⎝ ⎠ ⎝ ⎠

v 1
v 2
. . .
v n − 1
v n
v n + 1
. . .
v M

v 1
v 2
. . .
v n − 1

v − w = −

− ηexp i arg(v n )
0
. . . 0

. (2.207)

The choice of phase in the n th component of w the same as that of v n avoids the
possibility of subtracting nearly equal complex quantities in forming v − w . Since

u is a unit vector, we write

v − w

u = . (2.208)

| v − w |

Then,

i = n + 1 v i v ∗ i

M

2

(v n − w n )(v n − w n ) ∗ + | v − w | =

v n + ηexp i arg(v n ) v ∗ n + ηexp −

i arg(v n )

η

2 2

= + −| v n |

η

2

2 2 2

=| v n | + η + 2η | v n |+ −| v n |

2η η +| v n | ,

= (2.209)

and

⎝ ⎠

0
. . . 0
v n + ηexp i arg(v n )
v n + 1
. . .
v M

1
2η(η +| v n

u = . (2.210)

| )