Align the Truncated SVD from sklearn.decomposition and np.linalg.svd - numpy

Question

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I read an infomation in this book:
The matrix that is actually returned by TruncatedSVD is the dot product of the U andS matrices.
Then i try to just multiply U and Sigma:
US = U.dot(Sigma)
print("==>> US: ", US)
this time it produce the same result, just with sign flipping. So why Truncated SVD doesn't need multiplying VT ?
==========previous question===========
I am learning SVD, i found numpy and sklearn both provide some related APIs, then i try to use them to do dimensional reduction, below are the code:
import numpy as np
np.set_printoptions(precision=2, suppress=True)
A = np.array([
[1,1,1,0,0],
[3,3,3,0,0],
[4,4,4,0,0],
[5,5,5,0,0],
[0,2,0,4,4],
[0,0,0,5,5],
[0,1,0,2,2]])
U, s, VT = np.linalg.svd(A)
print("==>> U: ", U)
print("==>> VT: ", VT)
# create m x n Sigma matrix
Sigma = np.zeros((A.shape[0], A.shape[1]))
# populate Sigma with n x n diagonal matrix
square_len = min((A.shape[0], A.shape[1]))
Sigma[:square_len, :square_len] = np.diag(s)
print("==>> Sigma: ", Sigma)
n_elements = 2
U = U[:, :n_elements]
Sigma = Sigma[:n_elements, :n_elements]
VT = VT[:n_elements, :n_elements]
# reconstruct
B = U.dot(Sigma.dot(VT))
print("==>> B: ", B)
The output B is :
==>> B: [[ 0.99 1.01]
[ 2.98 3.04]
[ 3.98 4.05]
[ 4.97 5.06]
[ 0.36 1.29]
[-0.37 0.73]
[ 0.18 0.65]]
then this is sklearn code:
import numpy as np
from sklearn.decomposition import TruncatedSVD
A = np.array([
[1,1,1,0,0],
[3,3,3,0,0],
[4,4,4,0,0],
[5,5,5,0,0],
[0,2,0,4,4],
[0,0,0,5,5],
[0,1,0,2,2]]).astype(float)
svd = TruncatedSVD(n_components=2)
svd.fit(A) # Fit model on training data A
print("==>> right singular vectors: ", svd.components_)
print("==>> svd.singular_values_: ", svd.singular_values_)
B = svd.transform(A) # Perform dimensionality reduction on A.
print("==>> B: ", B)
its last output result is:
==>> B: [[ 1.72 -0.22]
[ 5.15 -0.67]
[ 6.87 -0.9 ]
[ 8.59 -1.12]
[ 1.91 5.62]
[ 0.9 6.95]
[ 0.95 2.81]]
As we can see, they produce different result (but i notice their singular values are the same, both are 12.48 9.51), how to make them same, does i misunderstand something ?

Answer 1

I think the correct way to perform a dimensionality reduction of the array A with np.linalg.svd is:
U, s, V = np.linalg.svd(A)
VT = V.T
B = A#VT[:,:n_elements]
Now B is:
array([[-1.72, 0.22],
[-5.15, 0.67],
[-6.87, 0.9 ],
[-8.59, 1.12],
[-1.91, -5.62],
[-0.9 , -6.95],
[-0.95, -2.81]])
That is exactly what you get from the TruncatedSVD, but with negative sign.