Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 137

time domain before any singular values are eliminated. As successive singular val-
ues are eliminated, starting with the smallest and working upward, the mean square
in the frequency domain appears to decrease monotonically until Parseval's rela-
tion (2.197) is closely obeyed. Thus, Palmer's procedure is to define the ratio R of
the right-hand side to the left-hand side of (2.197),

f ) 2 k =− N G k G ∗ k ( Δ

E g j g j

R = , (2.202)

and then to iterate on the number of singular values eliminated (NSVE) until the
ratio R is closest to unity.
The first step in the SVD of a Hermitian matrix, as described by Golub and
Kahan (1965), is the reduction of the matrix to upper bidiagonal form by a series
of Householder transformations. A detailed description of such annihilating trans-
formations as reflections is provided by Maron and Lopez (1991).
The basis of a Householder transformation is the matrix

2 u ⊗ u H P = I − , (2.203)

u H is the outer product of the vectors u and u H , where I is the unit matrix and u ⊗

⎝ ⎠

u 1 u ∗ 1 u 1 u ∗ 2 ··· u 1 u ∗ M

u 2 u ∗ 1 u 2 u ∗ 2 ··· u 2 u ∗ M

u H u ⊗ = , (2.204)

. . . . . . . . .

. . .

u M u ∗ 1 u M u ∗ 2 ··· u M u ∗ M

2

and the inner product of u H and u is u H

1, where u is a yet to
be specified unit vector. Thus, the outer product generates a matrix that is itself
Hermitian. Then,

· u = | u | =

I

u H

I

u H PP H = − 2 u ⊗ · − 2 u ⊗

u H u H u H = I − 4 u ⊗ + 4 u ⊗ · u ⊗

u H u H = I − 4 u ⊗ + 4 u ⊗

= I . (2.205)

The transformation matrix P is a unitary matrix for an arbitrary unit vector u .Now,
suppose the vector u is proportional to the di ff erence of two vectors, a vector v with

M components and a vector w , whose first n − 1 components are identical to those

of v , and whose n th component is − η, with

M

i = n v i v ∗ i

η = (2.206)