A square invertible G has the inverse G^{-1}. But forward operators are usually rectangular or rank-deficient, with no ordinary inverse. The Moore–Penrose pseudoinverse G^{+} is the one matrix that does the right thing in every case — it returns the best, simplest answer the data allows.

Its defining promise: \hat m = G^{+}d is always the least-squares, minimum-norm solution. Among all models that fit the data best, it picks the smallest.

Two regimes, one symbol

• Over-determined, full column rank (tall G): no exact solution, so G^{+} gives the least-squares fit — G^{+} = (G^{\mathsf T}G)^{-1}G^{\mathsf T}, the normal-equations operator.
• Under-determined, full row rank (wide G): infinitely many exact fits, so G^{+} gives the minimum-norm one — G^{+} = G^{\mathsf T}(GG^{\mathsf T})^{-1}.

When G is rank-deficient neither formula applies, because the relevant matrix is itself singular. The pseudoinverse still exists, but to build it we need a decomposition that exposes the rank directly — the SVD, the universal route to G^{+}.