[Python] Discrepancy in Signs Between scikit-learn PCA and Custom PCA Implementation

I’m implementing my own version of PCA and comparing it with scikit-learn's PCA. However, I’m noticing a discrepancy in the signs of the principal components.

Using scikit-learn

scaler = StandardScaler()
scaled_data = scaler.fit_transform(data)

pca_scaled = PCA(n_components=2)
pca_scaled.fit(scaled_data)

principal_components = pca_scaled.components_


Output:

[[ 0.70710678 0.70710678]
[-0.70710678 0.70710678]]


Custom PCA

def custom_PCA(data, num_components):
mean = np.mean(data, axis=0)
std = np.std(data, axis=0)
X_standardized = (data - mean) / std

cov_matrix = np.cov(X_standardized, rowvar=False)

eigenvalues, eigenvectors = np.linalg.eig(cov_matrix)

return eigenvectors


Output:

[[ 0.70710678 -0.70710678]
[ 0.70710678 0.70710678]]


Why the signs are flipped?

scikit-learn:
[[ 0.70710678 0.70710678]
[-0.70710678 0.70710678]]

Custom PCA:
[[ 0.70710678 -0.70710678]
[ 0.70710678 0.70710678]]

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