= Principal component analysis on a correlation matrix
{title2=$Z=D^{-1/2}(X-\mu),\quad\operatorname{Cov}(Z)=D^{-1/2}\Sigma D^{-1/2}$}
For positive marginal <variances>, standardizing each variable to <variance> one converts covariance-based <PCA> on $Z$ into correlation-based PCA on $X$. This makes the analysis insensitive to the original measurement units but changes the optimization criterion. Standardization gives small-variance variables the same marginal weight as large-variance variables; it is a modelling choice, not an automatic improvement.
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