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6.4. PROJECTION TO STETTER-STRUCTURE 91
This vector u has much resemblance with a Stetter vector with a structure corresponding
to (1, x 1 ,x 2 1,x 2 ,x 1 x 2 ,x 2 1x 2 ,x 2 2,x 1 x 2 2,x 2 1x 2 2) T where x 1 = −4 − 4i and
x 2 = −9 − 8i.
In polar coordinates the entries of the vector u can be written as u j = υ j e i·ψj for
j =1,...,N. This yields in this example:
⎛
υ =
⎜
⎝
1.000
4.000
29.428
13.928
72.173
380.938
140.911
816.486
4644.849
⎞
⎛
, ψ =
⎟
⎜
⎠
⎝
0.000
3.142
1.742
−2.774
1.502
−0.837
1.457
−0.901
3.024
⎞
. (6.29)
⎟
⎠
To project the vectors υ and ψ, the matrix W and the projection matrix P is
required. To construct the matrix W , the logarithmic transformation is applied to
the vector (1, x 1 ,x 2 1,x 2 ,x 1 x 2 ,x 2 1x 2 ,x 2 2,x 1 x 2 2,x 2 1x 2 2) T . This yields:
⎛
log
⎜
⎝
⎞
⎛
1
x 1
x 2 1
x 2
= log(x 1 )w 1 + log(x 2 )w 2 = log(x 1 )
⎟
⎜
⎠
⎝
x 1 x 2
x 2 1x 2
x 2 2
x 1 x 2 2
x 2 1x 2 2
0
1
2
0
1
2
0
1
2
⎞ ⎛
+ log(x 2 )
⎟ ⎜
⎠ ⎝
0
0
0
1
1
1
2
2
2
⎞
. (6.30)
⎟
⎠
The vectors w 1 and w 2 above constitute the matrix W of dimensions N × n =9× 2:
⎛
W =
⎜
⎝
0 0
1 0
2 0
0 1
1 1
2 1
0 2
1 2
2 2
⎞
. (6.31)
⎟
⎠
The symmetric projection matrix P of dimension N × N =9× 9 is constructed using