To generalize the concept of the inverse to non square matrix the pseudo inverse (also called Moore–Penrose inverse) can be used:
For tall matrices with full column rank: $$A^+ = (A^T \cdot A)^{-1} A^T$$
For wide matrices with full row rank: $$A^+ = A^T(A \cdot A^T)^{-1}$$
If A has not full rank(row or column) and is therefore singular the inverse of in the above equation is not defined anymore. To arrive at the damped Moore-Penrose pseudoinverse we can therefore introduce a damping term into the two equations above to make sure the inverse is defined $$A_d^+ = (A^T \cdot A + \lambda^2 \mathbb{I})^{-1} A^T$$ $$A_d^+ = A^T(A \cdot A^T + \lambda^2 \mathbb{I})^{-1}$$
Are the two formulas above equal for equal lambda and a general real matrix A? After trying out several examples it seems to me that this could be the case. If it is true, how would you formally proof this? $$A_d^+ = (A^T \cdot A + \lambda^2 \mathbb{I})^{-1} A^T= A^T(A \cdot A^T + \lambda^2 \mathbb{I})^{-1}$$