Exercise 4.10.2

$D=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 9\hfill & \hfill 25\hfill \\ \hfill 9\hfill & \hfill 0\hfill & \hfill 16\hfill \\ \hfill 25\hfill & \hfill 16\hfill & \hfill 0\hfill \end{array}\right]$and $d_1=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 9\hfill \\ \hfill 25\hfill \end{array}\right]$, according to equation (4), we have $G=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 9\hfill & \hfill 9\hfill \\ \hfill 0\hfill & \hfill 9\hfill & \hfill 25\hfill \end{array}\right]$

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

import numpy as np 
def find_X(G): 
   v, L = np.linalg.eig(G) 
   print(’Eigenvalues␣of␣G:␣\n’, v) 
   print(’Eigenvectors␣of␣G:␣\n’, L) 
 
   n = G.shape[0] 
 
   W = np.identity(n) 
   for i in range(n): 
       # Check orthogonality 
       for j in range(n): 
           d = np.dot(L[:,i], L[:,j]) 
           print(’Dot␣product␣of␣column␣’, i, ’␣with␣column␣’, j, ’␣is:␣’, d) 
       W[i,i]= v[i] 
   Gk = np.matmul(np.matmul(L, W), L.transpose()) 
   print(’The␣G␣with␣eigenvectors:␣\n’, Gk) 
   X = np.matmul(np.sqrt(W), L.transpose()) 
   # This recovers G as well 
   Gc = np.matmul(X.transpose(), X) 
   print(’The␣calculated␣G:␣\n’, Gc) 
   print(’The␣X␣is:␣\n’, X) 
   return X 
 
G= np.array([[0, 0, 0], [0, 9, 9], [0, 9, 25]]) 
X = find_X(G)

Eigenvalues of G:
 [29.04159458  4.95840542  0.        ]
Eigenvectors of G:
 [[ 0.          0.          1.        ]
 [ 0.40965605  0.91224006  0.        ]
 [ 0.91224006 -0.40965605  0.        ]]
Dot product of column  0  with column  0  is:  1.0
Dot product of column  0  with column  1  is:  0.0
Dot product of column  0  with column  2  is:  0.0
Dot product of column  1  with column  0  is:  0.0
Dot product of column  1  with column  1  is:  1.0
Dot product of column  1  with column  2  is:  0.0
Dot product of column  2  with column  0  is:  0.0
Dot product of column  2  with column  1  is:  0.0
Dot product of column  2  with column  2  is:  1.0
The G with eigenvectors:
 [[ 0.  0.  0.]
 [ 0.  9.  9.]
 [ 0.  9. 25.]]
The calculated G:
 [[ 0.  0.  0.]
 [ 0.  9.  9.]
 [ 0.  9. 25.]]
The X is:
 [[ 0.          2.20764686  4.91608482]
 [ 0.          2.03132847 -0.91220068]
 [ 0.          0.          0.        ]]

G= np.array([[9, 9], [9, 25]]) 
L = np.linalg.cholesky(G) 
X = L.transpose() 
print(’We␣find␣another␣X:␣\n’, X)

We find another X:
 [[3. 3.]
 [0. 4.]]