What is the MLE for centered Gaussian:$ S_y=\frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}$ or $S_y=\frac{1}{n}\boldsymbol{XX}^T$

Question

What is the Maximum Likelihood Estimator (MLE) for the centered ($\mu=0$) multivariate Gaussian?

My book gives the MLE as $$ S_y=\frac{1}{n}\boldsymbol{X}^T\boldsymbol{X} $$ while I've seen online resources derive the MLE as $$ S_y=\frac{1}{n}\boldsymbol{XX}^T $$

The context is to use PCA: $$\boldsymbol{X}=\boldsymbol{U\Sigma V}^T$$ and if the first definition is used, I would get the eigenvalues/eigenvectors of $\boldsymbol{U}$ while $\boldsymbol{V}$ if the second definition is used.