Simultaneous two-way centering and scaling

Scaling disturbs centering across the same mode, but not other modes 

Consider a two-way array X (I x J). If the array is scaled within the first mode, this can be expressed 

WPxX upon scaling. As l WPiX 0, the preprocessed matrix is no longer guaranteed to be centered. Offsets constant across the first mode, however, will still be removed, because 

Note also the interesting fact that if scaling is performed before centering, the result will be different. In that case, the original offsets will not be removed, but the data will be centered (yielding centered scores and residuals), because 

This holds, because unlike for Pxl, it does not hold that PXW1 is a zero vector in general. 

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