I have encountered the following equality

$\arg\max_{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{S}\right)=1}\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{A}\boldsymbol{S}\left(\boldsymbol{S}^{H}\boldsymbol{B}\boldsymbol{S}\right)^{-1}\right)=\arg\max_{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{S}\right)=1}\frac{\left|\boldsymbol{S}^{H}\boldsymbol{A}\boldsymbol{S}\right|}{\left|\boldsymbol{S}^{H}\boldsymbol{B}\boldsymbol{S}\right|}$

in the paper "Trace Ratio vs. Ratio Trace for Dimensionality Reduction".

However I can't seem to find its proof, how do we prove this equality?