 Principal component loading

Two conventions occur for principal-component loadings: the unit eigenvector ${\ell }_j$ defining a principal component, and the scaled vector $r_j=\sqrt{{\lambda }_j}{\ell }_j$. For principal component analysis on a correlation matrix, the entries of the latter are the correlations between each standardized variable and the component scaled to variance one. Indeed, $\mathrm{Cov}(Z,{\ell }_j^{T}Z)=C{\ell }_j={\lambda }_j{\ell }_j$ and $\mathrm{Var}({\ell }_j^{T}Z)={\lambda }_j$. Thus $\Vert r_j{\Vert }^2={\lambda }_j$, whereas $\Vert {\ell }_j\Vert =1$; reporting the convention avoids confusing component directions with correlations.