I know that the Tikhonov regularization of a linear system has an analytical solution given by:
\begin{equation} \hat{\mathbf{x}} = \mathrm{arg\;min}\left( \left| \mathbf{Ax} - \mathbf{b} \right|^{2} + \left| \mathbf{\Gamma x} \right|^{2} \right) = \left( \mathbf{A}^{\top}\mathbf{A} + \mathbf{\Gamma}^{\top}\mathbf{\Gamma} \right)^{-1} \mathbf{A}^{\top} \mathbf{b} \end{equation}
Is it possible to extend this idea to obtain an analytical solution for a system with an additional weighting? i.e.
\begin{equation} \hat{\mathbf{x}} = \mathrm{arg\;min}\left( \left| \mathbf{Ax} - \mathbf{b} \right|^{2} + \left| \mathbf{\Gamma x} \right|^{2} + \left| \mathbf{U x} \right|^{2} \right) = \; ? \end{equation}
Thanks for any help!