Given $(A^{T}A+L^{T}L)x=A^{T}b$ where $L=\alpha I$, I am asked to find $x$ in terms of $\alpha$ and singular values and vectors of $A$.
If I try using $A=U\Sigma V^{T}$ I eventually get to $(V\Sigma^{2}V^{T}+\alpha^{2}I)x=V\Sigma U^{T}b$ where I get stuck.
Since I am supposed to find $x$ in terms of singular values and vectors of $A$ I've also tried using $A=\sum_{i=1}^{r}\sigma_{i}\mathbf{u_{i}v_{i}}^{T}$ but I am not really sure how to proceed after substituting in the above relationship.
Any tips or help would be greatly appreciated.