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

I think there are still some important points for those who use SVD in Python/linalg library. Firstly, https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.svd.html is a good reference for SVD computation function.

Taking SVD computation as A= U D (V^T), For U, D, V = np.linalg.svd(A), this function returns V in V^T form already. Also D contains eigenvalues only, hence it has to be shaped into matrix form. Hence the reconstruction can be formed with

import numpy as np
U, D, V = np.linalg.svd(A)
A_reconstructed = U @ np.diag(D) @ V

The point is that, If A matrix is not a square but rectangular matrix, this won't work, you can use this instead

import numpy as np
U, D, V = np.linalg.svd(A)
m, n = A.shape
A_reconstructed = U[:,:n] @ np.diag(D) @ V[:m,:]

or you may use 'full_matrices=False' option in the SVD function;

import numpy as np
U, D, V = np.linalg.svd(A,full_matrices=False)
A_reconstructed = U @ np.diag(D) @ V