Principal component analysis on a correlation matrix ($Z=D^{-1/2}(X-\mu ),\mathrm{Cov}(Z)=D^{-1/2}\Sigma D^{-1/2}$)

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.