I believe the below background information is necessary to explain my question...
Background Information:
I have a square 2D matrix of variables that represent the interaction between any two positions in a protein's sequence. I then compare each column of interactions using Pearson's correlation and store that value in a new correlation matrix. This is shown graphically in the linked image.
The next step in this protocol is to take the partial correlation. For that I use the below equation where Where rxy-1 is the correlation coefficient for a given position (x, y) in the inverted correlation matrix, and rxx-1 and ryy-1 are the corresponding diagonal values.
On to the question itself, I found an alternative way of calculating partial correlations from this kind of data in this paper. It has two equations, shown below. As far as I can tell, the right equation is identical to the pc equation I have been using thus far, except with different notation, and it is shown as being equivalent to the left equation.
My question is: In converting from the right equation to the left equation, where is the third variable, C, coming from. A and B are equivalent to my X and Y position values, so how can I extrapolate a third C dimension from my 2D correlation matrix?
I have very little mathematical background, so don't hesitate to ask me for important information that I have overlooked. Thanks in advance for any help you all can render!!
Edit: Thanks to @GoldenGateBridge 's great explanation I think I now better understand the 3 variable partial correlation equation, at least the version used in the above paper. The right 2 var equation uses inverse covariance as its basis for PC calculation while the 3 var equation is calculated from the correlation matrix. I drew up a graphical representation of my understanding below (the numbers aren't accurate calculations, its just for example).
