Finding MLE by factor analyzing the correlation matrix

In the book "Applied Multivariate Statistical Analysis" written by Johnson and Wichern, they have mentioned that the MLEs ($\hat{L_z}$) are obtained from the the correlation matrix $R$ by inserting $R$ in place of $[(n-1)/n] \times S$ (where $S$ is the sample covariance matrix) in the likelihood. They also mention that the likelihood is appropriate for $S$, not for $R$, but surprisingly it is equivalent to obtaining MLE ($\hat{L}$) from $S$ and then setting $\hat{L_z}=\hat{V}^{-1/2}\hat{L}$.

Is there any proof of this? Can anyone help?