Given a linear system $Ax=b$, the pseudoinverse of $A$ is found as the matrix $A^+$ such that $x=A^+ b$ where $x$ solves the least squares problem $\min \| Ax - b \|^2 $ and $x \perp \mathcal{N}(A)$. That is, $x$ is the shortest vector in the solution space.

That is, find $x$ :
\begin{eqnarray}
\min \| x \| \text{ such that } x \text{ minimizes } \| Ax - b \|^2 .
\end{eqnarray}

This is similar to the regularization problem of minimizing
\begin{eqnarray}
 \| Ax - b \|^2 + \lambda \| x \|^2
\end{eqnarray}
I do not quite can get from one to the other? Are these two things equivalent?
Is there a way to post this as the same problem? Is there a connection?