Question

Given data \(d = \begin{bmatrix} d_{11} \\ d_{21} \end{bmatrix}\) and the kernel \(G = \begin{bmatrix} G_{11} & G_{12} & G_{13} \\ G_{21} & G_{22} & G_{23} \end{bmatrix}\), which one of the following expressions correctly represents the generalized linear inverse formula for the model, \(m\), satisfying \(d = Gm\)?

• A. \((G^TG)^{-1}G^T\)
• B. \(G^T(GG^T)^{-1}\)
• C. \((GG^T)^{-1}G\)
• D. \(GG^TG^{-1}\)

Hint:

With 2 data and 3 model parameters the system is underdetermined; minimize ||m||^2 subject to Gm=d using a Lagrange multiplier to get m = G^T(GG^T)^{-1}d.

Answer 1

The Correct Option is B

Here \(d\) is \(2 \times 1\) (two data values) and \(G\) is \(2 \times 3\) (three model parameters, two equations), so the system \(d = Gm\) is UNDERDETERMINED -- fewer equations than unknowns, meaning infinitely many exact solutions \(m\) satisfy \(Gm = d\). The standard way to pick a unique, well-behaved solution from this infinite family is the minimum-norm solution: among all \(m\) satisfying \(Gm=d\), choose the one with the smallest \(\|m\|^2 = m^Tm\).

Set up the constrained minimisation with a Lagrange multiplier vector \(\lambda\):

\[ L = m^Tm - \lambda^T(Gm - d) \]

Differentiating with respect to \(m\) and setting it to zero:

\[ 2m - G^T\lambda = 0 \quad\Rightarrow\quad m = \tfrac{1}{2}G^T\lambda \]

Absorbing the factor of 2 into \(\lambda\), write \(m = G^T\lambda\). Substitute into the constraint \(Gm = d\):

\[ G(G^T\lambda) = d \quad\Rightarrow\quad (GG^T)\lambda = d \quad\Rightarrow\quad \lambda = (GG^T)^{-1}d \]

(this inverse exists because \(GG^T\) is a \(2\times2\) matrix, which is invertible whenever \(G\) has full row rank -- the underdetermined case). Substituting back:

\[ m = G^T(GG^T)^{-1}d \]

so the generalized (minimum-norm) inverse operator that maps data to model is \(G^T(GG^T)^{-1}\), matching option (B). This is the standard underdetermined generalized inverse, complementary to the overdetermined least-squares inverse \((G^TG)^{-1}G^T\) used when there are more equations than unknowns (option A, not applicable here since \(G^TG\) would be a singular \(3\times3\) matrix for only 2 independent rows).

\(\boxed{G^{-g} = G^T(GG^T)^{-1}}\)