Understanding a "computationally convenient uniqueness condition" on loadings in factor analysis In  "Applied Multivariate Statistical Analysis"  by Johnson and Wichern, the authors mention a "computationally convenient uniqueness condition" $$L^T\psi^{-1}L=\Delta,$$ where $\Delta$ is a diagonal matrix. Here, $L$ is the factor loadings matrix and $\psi$ is vector of specific variances.
I cannot understand how this condition removes the multiple choices of $L$ and makes it unique. Can anyone help?
 A: Factor loadings $L$ and the diagonal matrix of specific variances $\Psi$ are chosen to approximate the covariance/correlation matrix $\Sigma$ of your data: $$\Sigma \approx LL^\top + \Psi.$$ We can immediately see that $L$ is not uniquely defined because any $LR$ where $R$ is a rotation matrix would work equally well: $$LR(LR)^\top = LRR^\top L^\top = LL^\top.$$
Your textbook introduces a constraint $$L^\top\Psi^{-1}L = \Delta.$$ If we can find a unique $R$ that would make this constraint true, then it would mean that this constraint fixes the choice of $L$. Let's try. Plugging $LR$ instead of $L$ into the constraint equation, we get $$R^\top L^\top \Psi^{-1} L R = \Delta.$$ Multiplying by $R$ from the left and by its transpose from the right, we obtain $$L^\top \Psi^{-1} L = R\Delta R^\top,$$ which is simply an eigendecomposition of $L^\top \Psi^{-1} L$. So $R$ is given by the matrix of eigenvectors of $L^\top \Psi^{-1} L$. QED.
PS. By the way, do the authors explain why this constraint is "computationally convenient"?
