Python manual SVD only working for some matrices - how to stabilize it?

Question

I use the following code for a manual single value decomposition using numpy. Depending on the array I choose it sometimes works out well and I can verify the svd and sometimes it does not work out straight aways and requires a sign flip.

Code

import numpy as np

array = np.array([[-3,3,6],
                  [3,8,7]])  # here you can choose any matrix values. Some matrices work, some don't.

# left singular vectors
AAT = np.matmul(array, array.T)
eAAT_values, eAAT_vectors = np.linalg.eig(AAT)
idx = eAAT_values.argsort()[::-1]      #sorting: largest singular values first
eAAT_values = eAAT_values[idx]
eAAT_vectors = eAAT_vectors[:,idx]
SL = eAAT_vectors
# Some arrays require an additional line:
# SL[:,0] = -SL[:,0]


# right singular vectors
ATA = np.matmul(array.T,array)
eATA_values, eATA_vectors = np.linalg.eig(ATA)
idx = eATA_values.argsort()[::-1]     #sorting: largest singular values first
eATA_values = eATA_values[idx]
eATA_vectors = eATA_vectors[:,idx]
SR = eATA_vectors.T


# singular values
S0 = np.zeros(np.shape(array))
np.fill_diagonal(S0, np.sqrt(eATA_values), wrap=True)

# verifying. Expected proof == array
Proof = np.matmul(SL,np.matmul(S0,SR))     # works out with some arrays with others it does not

# Python linalg.svd works consistently
U, S, Vt = np.linalg.svd(array) 
Sm = np.zeros(np.shape(array))
np.fill_diagonal(Sm, S, wrap=True)
proof2 = np.matmul(U, np.matmul(Sm,Vt))    # always works out

The array in the code above works well. Contrary examples where the svd does not work out without an additional sign flip are presented here.

array_not_working1 = np.array([[4,5,9],
                               [3,2,6]])

array_not_working2 = np.array([[3,-1,4],
                               [1,5,9]])

How can I stabilize the outcome or determine when a signflip of the first eigenvector from eAAT_vectors is necessary?

Another working array without an additional sign flip:

array = np.array([[7,-12,2], [-4,3,1]]).

Answer 1

This looks the same as I implemented an SVD, but sometimes it is not possible to restore the original matrix

You may look at my answer and the comments below.