\displaystyle\begin{aligned}  \Vert \mathbf{u}\mathbf{v}^T\Vert^2_F&=\text{trace}\left((\mathbf{u}\mathbf{v}^T)^T(\mathbf{u}\mathbf{v}^T)\right)=\text{trace}(\mathbf{v}\mathbf{u}^T\mathbf{u}\mathbf{v}^T)\\  &=\text{trace}(\mathbf{u}^T\mathbf{u}\mathbf{v}^T\mathbf{v})=\text{trace}(\Vert\mathbf{u}\Vert^2\Vert\mathbf{v}\Vert^2)=\Vert\mathbf{u}\Vert^2\Vert\mathbf{v}\Vert^2.  \end{aligned}