= Principal component loading
{title2=$r_j=\sqrt{\lambda_j}\ell_j$}
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, $\operatorname{Cov}(Z,\ell_j^TZ)=C\ell_j=\lambda_j\ell_j$ and $\operatorname{Var}(\ell_j^TZ)=\lambda_j$. Thus $\|r_j\|^2=\lambda_j$, whereas $\|\ell_j\|=1$; reporting the convention avoids confusing component directions with correlations.
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