For the solution of $Ax = b$, where $A$ is a square matrix, what is the difference between these two regularized solutions:

 - $x = (A + \alpha I)^{-1}b$ -- coressponding to eq.3 below 
 - $x = (A^TA + \alpha I)^{-1}A^Tb$ -- corresponding to eq.2 below with $K(w) = A, K(-w) = A^T$


In the following reference (equation 3):

https://www.sciencedirect.com/science/article/abs/pii/0041555382900945#

[![enter image description here][1]][1]

for positive symmetric difference kernels (for example a Gaussian kernel) the first form of regularization is promoted as a simplified Tikhonov regularization (we usually use $p = 0$ case for $M(w) = I$).

**Is the simplification just an approximation or is it equivalent to the second form above?** Basically I am trying to understand the connection between discretized versions that we use in statistics and the continuous versions that are used in integral equations.


  [1]: https://i.stack.imgur.com/TPntJ.jpg