Relative Gain Array of a singular matrix I am a masters student in controls and would like to get insight into the concept of relative gain array for multivariable feedback control. In general what I have come across from the book on the same topic by Skogestad and Postlethwaite, relative gain array element $\lambda_{i,j}$ is given by $G_{i,j}G^{-1}_{j,i}$, where G is a square matrix.
But how to find it if the matrix G is not invertible? In this case the matrix is
$G = \left[\begin{matrix} 1 & 1 \\ 1 & 1\end{matrix}\right]$. Is there a different approach I can take, a different definition perhaps to calculate the RGA?
 A: For a $2\times 2$ matrix, we have a closed-form expression for the inverse 
$$X^{-1}=\frac{I\,{\rm tr}(X)-X}{\det(X)}$$
Create a new matrix by perturbing your $G$ matrix in the direction of the identity matrix
$$\eqalign{
 X &= G+hI \cr
 \det(X) &= (1+h)^2-1 = 2h+h^2 \cr
{\rm tr}(X) &= 2(1+h) \cr
 X^{-1} &= \frac{2(1+h)I-(G+hI)}{2h+h^2} \cr
}$$
The RGA can be calculated using the Hadamard (aka element-wise) product 
$$\eqalign{
 R &= X\circ X^{-T} = X\circ X^{-1} = (G+hI)\circ X^{-1} \cr\cr
   &= \frac{2(1+h)(G+hI)\circ I-(G+hI)\circ(G+hI)}{2h+h^2} \cr
   &= \frac{2(1+h)^2I-\big(G+hI+h(1+h)I\big)}{2h+h^2} \cr
   &= \frac{2(1+2h+h^2)I-\big(G+(2h+h^2)I\big)}{2h+h^2} \cr
   &= \frac{(2+2h+h^2)I-G}{2h+h^2} \cr
   &= I + \frac{2I-G}{2h+h^2} \approx I + \frac{2I-G}{2h} \cr\cr
}$$
If you redo the perturbation analysis in the direction of the counter-identity $K$, you obtain a similar expression 
$$\eqalign{
 R &\approx K + \frac{2K-G}{2h} \cr
}$$
So you can find expressions for the RGA near this value of $G$, if that helps.
A: The concept of Moore-Pensrose pseudoinverse can be used to approximate for some properties of inverse in this case.
