Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 139

If we then perform the unitary transformation P on the vector v ,wefindthat

u H

· v = v −

2 u

⊗ · v

P

η

+ ηv n exp − i arg (v n )
2η(η +| v n |

2

= v − 2 u

)

)
2η(η +| v n

2η(η +| v n |

= v − u

| )

= v − ( v − w ) = w . (2.211)

Taking the vector v to be a column vector of an M × M matrix, elements below
v n can be brought to zero by a unitary Householder transformation. Similarly, if
v T is a row vector, the superscript T indicating the transpose of the corresponding
column vector, then on taking the transpose of the product P · v ,wehave

I

u T v T P T v T P ∗ = v T 2 u ∗ ⊗ v T 2 v T u ∗ ⊗ u T · = · · − = − ·

v T

w T v T w T = − − = . (2.212)

Thus, the elements of the row vector beyond v n can be brought to zero by post-
multiplying by the complex conjugate of the Householder matrix. By alternating
column and row operations, it is found that previous annihilations are una ected,

and the whole matrix can be reduced to upper bidiagonal form by unitary trans-
formations.
We illustrate the succession of column and row operations required to reduce a
complex matrix to upper bidiagonal form, with the following 5

ff

× 5matrix:

real part of matrix is,
0.28488D+03 -0.12135D+03 -0.18145D+03 0.27585D+03 -0.53595D+02
-0.12135D+03 0.28488D+03 -0.12135D+03 -0.18145D+03 0.27585D+03
-0.18145D+03 -0.12135D+03 0.28488D+03 -0.12135D+03 -0.18145D+03
0.27585D+03 -0.18145D+03 -0.12135D+03 0.28488D+03 -0.12135D+03

-0.53595D+02 0.27585D+03 -0.18145D+03 -0.12135D+03 0.28488D+03

imaginary part of matrix is,
0.00000D+00

0.25772D+03 -0.21954D+03 -0.70624D+02 0.27956D+03

-0.25772D+03 0.00000D+00 0.25772D+03 -0.21954D+03 -0.70624D+02

0.21954D+03 -0.25772D+03 0.00000D+00 0.25772D+03 -0.21954D+03

0.70624D+02 0.21954D+03 -0.25772D+03 0.00000D+00 0.25772D+03

-0.27956D+03 0.70624D+02 0.21954D+03 -0.25772D+03 0.00000D+00.

The first step in the reduction to upper bidiagonal form is to pre-multiply by the
unitary Householder matrix constructed to annihilate the elements below the diag-
onal in the first column, then to post-multiply by the unitary Householder matrix
constructed to annihilate the elements above the superdiagonal in the first row. The
matrix then takes the form