# ## MIT License

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# In[1]:
import numpy as np
import numpy.linalg as la
# In[2]:
np.random.seed(15)
n = 5
A = np.random.randn(n, n)
# Now compute the eigenvalues and eigenvectors of $A^TA$ as `eigvals` and `eigvecs` using `la.eig` or `la.eigh` (symmetric):
# In[11]:
eigvals, eigvecs = la.eigh(A.T @ A)
# In[12]:
eigvals
# Eigenvalues are real and non-negative. Coincidence?
# In[13]:
eigvecs.shape
# Check that those are in fact eigenvectors and eigenvalues:
# In[14]:
B = A.T @ A
B - eigvecs @ np.diag(eigvals) @ la.inv(eigvecs)
# `eigvecs` are orthonormal! (Why?)
#
# Check:
# In[15]:
la.norm(eigvecs.T @ eigvecs - np.eye(n))
# ------
# Now piece together the SVD:
# In[16]:
Sigma = np.diag(np.sqrt(eigvals))
# In[17]:
V = eigvecs
# In[18]:
U = A @ V @ la.inv(Sigma)
# Check orthogonality of `U`:
# In[19]:
U @ U.T - np.eye(n)
# In[ ]: