How centering and scaling data affects PCA

As with many of the transformations that we have worked with previously in this text, the scaling of features tends to matter a great deal to the transformations. PCA is no different. Previously, we mentioned that the scikit-learn version of PCA automatically centers data in the prediction phase, but why doesn't it do so at the fitting time? If the scikit-learn PCA module goes through the trouble of centering data in the predict method, why doesn't it do so while calculating the eigenvectors? The hypothesis here is that centering data doesn't affect the principal components. Let's test this:

1. Let's import out StandardScaler module from scikit-learn and center the iris dataset:

# import our scaling module
 from sklearn.preprocessing import StandardScaler
 # center our data, not a full scaling
 X_centered = StandardScaler(with_std=False).fit_transform(iris_X)

 X_centered[:5,]

 array([[-0.74333333, 0.446 , -2.35866667, -0.99866667], [-0.94333333, -0...