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W 4 gives: 1
The eigenstructure analysis of matrix
D0:14, 2
D 0:16,
3 D0:02, with associated eigenvectors v W 4
1
D Œ0:7071; 0:7071; 0 T , v W 4
2
D
Œ0:6667; 0:6667; 0:3333 T , and v W 3 D Œ0:2357; 0:2357; 0:9428 T .
The eigenstructure analysis of matrix W 5 gives: 1 D0:16, 2 D 0:1649,
3 D0:0049, with associated eigenvectors v W 5
1
D Œ0:7071; 0:7071; 0 T , v W 5
2
D
Œ0:6969; 0:6969; 0:1691 T , and v W 3 D Œ0:1196; 0:1196; 0:9856 T .
The eigenstructure analysis of matrix W 6 gives: 1 D0:1613, 2 D0:0093,
3 D 0:1706, with associated eigenvectors v W 6
1
D Œ0:6957; 0:7126; 0:0905 T ,
v W 6
2 D Œ0:2353; 0:1071; 0:9660 T , and v W 3 D Œ0:6787; 0:6933; 0:2421 T .
The eigenstructure analysis of matrix W 7 gives: 1 D0:1613, 2 D0:0093,
3 D0:1706, with associated eigenvectors v W 7
1
D Œ0:7126; 0:6957; 0:0905 T ,
v W 7
2 D Œ0:1071; 0:2353; 0:9660 T , and v W 3 D Œ0:6993; 0:6787; 0:2421 T .
The eigenstructure analysis of matrix W 8 gives: 1 D0:1600, 2 D 0:1780,
3 D0:0180, with associated eigenvectors v W 8
1
D Œ0:7071; 0:7071; 0 T , v W 8
2
D
Œ0:6739; 0:6739; 0:3029 T , and v W 8
3
D Œ0:2142; 0:2142; 0:9530 T .
Remark 2. In neural structures with weights that follow Schrödinger's equation
with zero or constant potential (see the analysis in [ 165 ]), the probability to recall
the pattern v W k ;iD 1; ;8; k D 1; ;3 is proportional to the membership
i of the matrix
W i , i.e. P / . W i /. The superimposing matrices
W i 's describe a
distributed associative memory [ 96 ].
Remark 3. The difference of the neural structures that follow the quantum harmonic
oscillator (QHO) model comparing to neural structures that follow Schrödinger's
equation with zero or constant potential is that convergence to an attractor is
controlled by the drift force imposed by the harmonic potential [ 124 ]. Convergence
to an attractor, as the results of Chap. 8 indicate, can be steered through the drift
force, which in turn is tuned by the parameters a and b of Eq. ( 8.14 ).
2. Unitarity of the rotation operators
Here the analysis of second part of Sect. 10.2 will be verified. Matrix R of
Eq. ( 10.10 ) which performs a rotation from the basis defined by the eigenvectors
v . W 1 /
1
; v . W 1 /
2
; v . W 1 3 , to the basis defined by the vectors v . W 2 /
; v . W 2 /
2
; v . W 2 /
3
is calculated
1
as follows: v . W 1 /
1
D Œ0:7071; 0:7071; 0 T , v . W 1 /
2
D Œ0:6941; 0:6941; 0:1908 T and
v W 1
3
D Œ0:1349; 0:1349; 0:9816 T , while v . W 2 /
1
D Œ0:6921; 0:7143; 0:1041 T ,
v . W 2 /
2
D Œ0:2648;0:1176; 0:9972 T , and v . W 2 /
3
D Œ0:6715; 0:6900; 0:2701 T .
The rotation matrix R is given by
0
@
1
A D
0
1
v . W 1 /
1
v . W 2 /
1
v . W 1 /
1
v . W 2 /
2
v . W 1 /
1
v . W 2 /
3
0:9945 0:1041 0:0131
0:0045 0:0752 0:9966
0:1052 1:0304 0:0815
v . W 1 /
2
v . W 2 /
1
v . W 1 /
2
v . W 2 /
2
v . W 1 /
2
v . W 2 /
3
@
A
R D
v . W 1 /
3
v . W 2 /
1
v . W 1 /
3
v . W 2 /
2
v . W 1 /
3
v . W 2 /
3
(10.14)
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