Exercise 4.10.3

$D=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$and $d_1=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$, according to equation (4), we have $G=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill & \hfill \frac{1}{2}\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]$

The points lie in $R^3$ here.

We solve the $X$ using eigenvalue/eigenvectors of $G$ as below.

#G= np.array([[0, 0, 0, 0], [0, 1, 0.5, 0.5], [0, 0.5, 1, 0.5], [0, 0.5, 0.5, 1]]) 
# Since numpy.linalg.eig() doesn’t return orthogonal eigenvector matrix, so we use cholesky here 
G= np.array([[1, 0.5, 0.5], [0.5, 1, 0.5], [0.5, 0.5, 1]]) 
L = np.linalg.cholesky(G) 
X = L.transpose() 
print(’The␣final␣X:␣\n’, X)

The final X:
 [[1.         0.5        0.5       ]
 [0.         0.8660254  0.28867513]
 [0.         0.         0.81649658]]
    

1/np.sqrt(7)

0.3779644730092272