# coding: utf-8

# # Computing the SVD

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import numpy as np
import numpy.linalg as la


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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):

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eigvals, eigvecs = la.eigh(A.T.dot(A))


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eigvals


# Eigenvalues are real and positive. Coincidence?

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eigvecs.shape


# Check that those are in fact eigenvectors and eigenvalues:

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B = A.T.dot(A)
B - eigvecs.dot(np.diag(eigvals)).dot(la.inv(eigvecs))


# `eigvecs` are orthonormal! (Why?)
# 
# Check:

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eigvecs.T.dot(eigvecs) - np.eye(n)


# ------
# Now piece together the SVD:

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Sigma = np.diag(np.sqrt(eigvals))


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V = eigvecs


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U = A.dot(V).dot(la.inv(Sigma))


# Check orthogonality of `U`:

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U.dot(U.T) - np.eye(n)