The relative gain array (RGA) is a classical widely-used method for determining the best input-output pairings for multivariable process control systems. It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.
Definition.
Given a linear time-invariant (LTI) system represented by a nonsingular matrix formula_1, the relative gain array (RGA) is defined as 
formula_2
where formula_3 is the elementwise Hadamard product of the two matrices, and the transpose operator (no conjugate) is necessary even for complex formula_1. Each formula_5 element formula_6 gives a scale invariant (unit-invariant) measure of the dependence of output formula_7 on input formula_8.
Properties.
The following are some of the linear-algebra properties of the RGA:
The second property says that the RGA is invariant with respect to nonzero scalings of the rows and columns of formula_1, which is why the RGA is invariant with respect to the choice of units on different input and output variables. The third property says that the RGA is consistent with respect to permutations of the rows or columns of formula_1.
Generalizations.
The RGA is often generalized in practice to be used when formula_1 is singular, e.g., non-square, by replacing the inverse of formula_1 with its Moore–Penrose inverse (pseudoinverse). However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used. 
References.
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