Principal component loading ($r_j=\sqrt{{\lambda }_j}{\ell }_j$)

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, $\mathrm{Cov}(Z,{\ell }_j^{T}Z)=C{\ell }_j={\lambda }_j{\ell }_j$ 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.

In principal component analysis, the matrix whose columns are the chosen orthonormal eigenvectors of a covariance matrix, ordered by decreasing eigenvalues. Under this convention the principal component scores of centered data are $Z=X_{c}V$. Some literature instead scales the vectors by square roots of eigenvalues; the convention should be stated explicitly.