Using Numpy (np.linalg.svd) for Singular Value Decomposition

Question

Im reading Abdi & Williams (2010) \"Principal Component Analysis\", and I\'m trying to redo the SVD to attain values for further PCA.

The article states that followi

Answer 1

TL;DR: numpy's SVD computes X = PDQ, so the Q is already transposed.

SVD decomposes the matrix X effectively into rotations P and Q and the diagonal matrix D. The version of linalg.svd() I have returns forward rotations for P and Q. You don't want to transform Q when you calculate X_a.

import numpy as np
X = np.random.normal(size=[20,18])
P, D, Q = np.linalg.svd(X, full_matrices=False)
X_a = np.matmul(np.matmul(P, np.diag(D)), Q)
print(np.std(X), np.std(X_a), np.std(X - X_a))

I get: 1.02, 1.02, 1.8e-15, showing that X_a very accurately reconstructs X.

If you are using Python 3, the @ operator implements matrix multiplication and makes the code easier to follow:

import numpy as np
X = np.random.normal(size=[20,18])
P, D, Q = np.linalg.svd(X, full_matrices=False)
X_a = P @ diag(D) @ Q
print(np.std(X), np.std(X_a), np.std(X - X_a))
print('Is X close to X_a?', np.isclose(X, X_a).all())