The solution to Tikhonov Regularization is
$$x=(A^HA+\sigma^2\_{min}I)^{-1}A^Hb$$
where $\sigma^2\_{min}$ is the minimum of the singular values of $A$.


Then we apply $SVD$ to $A$ such that,
$$A=U\Sigma V^H$$


then the solution is,
$$x=(V\Sigma^2 V^H+\sigma^2\_{min} I)^{-1}V\Sigma U^Hb$$
But on the textbook, it says that the solution could be simplified such as
$$x=V(\Sigma^2+\sigma^2\_{min} I)^{-1}\Sigma U^Hb$$


I cannot find the solution above myself. I guess Woodbury Matrix Indentity should be helpful...


Thanks! 



How to derive the solution of Tikhonov Regularization via SVD
No need for the Woodbury identity. Just replace $I$ with $VV^H$, and factor out the $V$s.


