Factor rotation ($L\mapsto LQ,F\mapsto Q^{T}F$)

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.

A representation of a factor solution allowing correlated latent factors. If $X=LF+ϵ$ and $H$ is invertible, replacing $L$ by $LH$ and $F$ by $H^{-1}F$ preserves the shared covariance when the factor covariance matrix changes accordingly. Report factor correlations as well as loadings.

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.