Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 142

d ∗ 1 / | Choosing u 11 , making the first diagonal element real

and positive. Notice that the pre-multiplying matrix is unitary, since multiplying by
its Hermitian transpose gives the unit matrix. We next post-multiply this result as
⎝

= d 1 | gives u 11 d 1 = | d 1 |

⎠ ⎝ ⎠ = ⎝ ⎠

| d 1 | u 11 e 1 0 100

0 v 22 0
001

| d 1 | v 22 u 11 e 1 0

0 d 2 e 2 0 v 22 d 2 e 2 . (2.215)

0 0 d 3 0 0 d 3

u ∗ 11 · e ∗ 1 / |

Taking v 22 = e 1 | gives v 22 u 11 e 1 = | e 1 | . Once again, the post-multiplying

matrix is unitary, since v 22 v ∗ 22 =

1. We then pre-multiply as

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

100
0 u 22 0
001

| d 1 | | e 1 | 0 | d 1 | | e 1 | 0

=

0 v 22 d 2 e 2 0 u 22 v 22 d 2 u 22 e 2 . (2.216)

0 0 d 3 0 0 d 3

= v ∗ 22 · d ∗ 2 / | Taking u 22 d 2 | gives u 22 v 22 d 2 = | d 2 | . The pre-multiplying matrix is

unitary since u 22 u ∗ 22 = 1. Again, we post-multiply as

⎝ ⎠ ⎝ ⎠ = ⎝ ⎠

| d 1 | | e 1 | 0 100

010
00v 33

| d 1 | | e 1 | 0

0 | d 2 | u 22 e 2

00 d 3

0 | d 2 | v 33 u 22 e 2

00v 33 d 3

. (2.217)

u ∗ 22 · e ∗ 2 / |

Taking v 33 = e 2 | gives v 33 u 22 e 2 = | e 2 | . The post-multiplying matrix is

unitary since v 33 v ∗ 33 =

1. Finally, we pre-multiply as

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

10 0
01 0
00 u 33

| d 1 | | e 1 | 0 | d 1 | | e 1 | 0

0 | d 2 | | e 2 |

00v 33 d 3

= 0 | d 2 | | e 2 |

00 u 33 v 33 d 3

. (2.218)

= v ∗ 33 · d ∗ 3 / | Taking u 33 d 3 | gives u 33 v 33 d 3 =| d 3 | . The pre-multiplying matrix is unitary

since u 33 u ∗ 33 = 1. The transformed real, positive matrix is then

⎝ ⎠

| d 1 | | e 1 |

0

0 | d 2 | | e 2 |

00

. (2.219)

| d 3 |

The complete transformation is seen to depend on the determination of the unit
length phasors u 11 , v 22 , u 22 , v 33 , u 33 . These can be computed recursively as

d ∗ 1 e ∗ 1 d ∗ 2

u ∗ 11 , u 22 = v ∗ 22

u 11 = , 22 = ,

| d 1 | | e 1 | | d 2 |

e ∗ 2 d ∗ 3

u ∗ 22 = v ∗ 33 v 33 = , u 33 . (2.220)

| e 2 | | d 3 |

The ultimate objective of singular value decomposition is to factor the matrix
C into the triple product (2.199). Thus far, our reduction of the matrix to upper