Orthogonal factor model ($X=\mu +LF+ϵ,\mathrm{Cov}(X)=L{L}^T+\Psi$)

The latent factor vector has mean zero and covariance $I_q$, the specific errors have mean zero and diagonal covariance $\Psi$, and the two are uncorrelated. Thus the off-diagonal covariance is explained by shared factors. Normality additionally gives a likelihood model, but the covariance identity needs only these second-moment assumptions. Positive diagonal entries of $\Psi$ describe variation not explained by the factors. Unlike principal component analysis, the model separates shared and specific variation rather than only maximizing total projected variance.

An orthogonal rotation of factors leaves $L{L}^T$ unchanged and hence preserves the model covariance. It proves rotational nonidentifiability of the unrestrained factor loadings. Choose an orientation for interpretability, for example varimax rotation, or impose identifying restrictions in a confirmatory model. An oblique transformation permits correlated factors: if $L$ becomes $LH$, their covariance becomes $H^{-1}{H}^{-T}$, preserving the same shared covariance.

An orthogonal factor rotation maximizing, over factor axes, the variance across variables of their squared loadings. It seeks an interpretable pattern with some large and many small loadings on each factor without changing the covariance fit.

The communality is the variance of the shared-factor contribution to variable $i$. In an orthogonal factor model, $\mathrm{Var}(X_i)=h_i^2+{\psi }_i$, where ${\psi }_i$ is its specific variance. For standardized variables, $h_i^2\le 1$. It measures variance shared through the modeled factors, not variance in a chosen principal component.

A factor loading is the coefficient of latent factor $F_j$ in observed variable $X_i$. In the orthogonal factor model, $\mathrm{Cov}(X_i,F_j)=L_{ij}$. If $X_i$ has variance one, this is also its correlation with the variance-one factor. Its sign and size depend on the factor-coordinate convention; factor rotation preserves the implied covariance while changing individual loadings.