Given an augmented matrix $$M = [A \mid X] = \left[\begin{array}{cc|c} a & b & x\\ c & d & y \end{array}\right],$$
there's an associated augmented matrix defined like so $$M' = [A' \mid X'] = \left[\begin{array}{cc|c} \dfrac{ax}{ax+cy} & \dfrac{cy}{ax+cy} & ax+cy\\ \dfrac{bx}{bx+dy} & \dfrac{dy}{bx+dy} & bx+dy \end{array}\right],$$
as long as neither of the denominators vanish. It's probably best to assume $x,y>0$ and $a,b,c,d \geq 0$ together with $a + b = 1$ and $c + d = 1$ to prevent division by zero.
Anyway, this function $M \mapsto M'$ has good theoretical properties; it's analytic, involutive, and so long as $A$ is row-stochastic and $X$ is column-stochastic, it follows that $A'$ is row stochastic and $X'$ is column stochastic. Going beyond theoretical considerations, I'll note that this transform shows up in a lot of real-world applications, whenever you're trying to perform binary classification based on an imperfect test.
Example.
If
- $M(1,1)$ is the probability of the test coming up positive assuming you have the disease
- $M(1,2)$ is the probability of the test coming up negative assuming you have the disease
- $M(1,3)$ is the probability of having the disease
- $M(2,1)$ is the probability of the test coming up positive assuming you don't have the disease
- $M(2,2)$ is the probability of the test coming up negative assuming you don't have the disease
- $M(2,3)$ is the probability of not having the disease
then
- $M'(1,1)$ is the probability you have the disease assuming that the test comes up positive
- $M'(1,2)$ is the probability you don't have the disease assuming that the test comes up positive
- $M'(1,3)$ is the probability that the test comes up positive
- $M'(2,1)$ is the probability you have the disease assuming that the test comes up negative
- $M'(2,2)$ is the probability you don't have the disease assuming that the test comes up negative
- $M'(2,3)$ is the probability that the test comes up negative
It's straightforward to generalise the function $(M \mapsto M')$ so that it can be applied to any augmented square matrix in which the denominators don't vanish.
Question. Does this transform (and/or the matrix it produces) have a name?
Addendum 5th Septemeber, 2020. I think I'll call it the conditional dual for now. I remain very interested in this notion, and would love to hear from anyone who knows more about it.