The Moore-Penrose pseudoinverse $A^{\dagger}$ generalizes the matrix inverse:

• If $A$ is square and invertible: $A^{\dagger} = A^{-1}$
• If $A$ has full column rank: $A^{\dagger} = (A^TA)^{-1}A^T$ (left inverse)
• If $A$ has full row rank: $A^{\dagger} = A^T(AA^T)^{-1}$ (right inverse)
• For any matrix: computed via SVD as $A^{\dagger} = V\Sigma^{\dagger}U^T$

Applications:

• Solving least squares problems: $\min \|Ax - b\|$
• Finding minimum norm solutions to $Ax = b$
• Computing best-fit approximations