PCA performs a linear transformation on a data set to obtain a new data set, this time with eigenvector basis and eigenvalue loadings. If Z is our data set, P our linear transformation, and Y our new data set with eigenvector basis, the transformation is:
P*Z = Y .
but, quiet often Z is not our original data set. If the units of the variables are different, we must perform some sort of standardization/normalization transformation, say K, so that they are comparable. If K is our transformation matrix, then to obtain the new, standardized data set, we perform:
K*X = Z
My question is, what are the loadings in terms of our original data set X? In more practical terms, if b1 is the first measurement type of Z, PC1 might look like..
Where the values of b1->bn are given by P. But what is PC1 in terms of the original data set X, with measurements a1->an?
These should come from some new linear transformation P_new. I would assume we can use basic matrix algebra to figure this out, but I want to make sure I'm 100% correct here:
if K*X = Z,
and P_old*Z = Y, we're searching for:
P_new*X = Y
we should be able to use...
P_new*X = Y
or P_new = Y*X^-1 ?
Is this correct? What if X is not square (and it most certainly is not, there are almost always more trials than variables in any good data set)? Will computational solutions make sense?
I've searched through 23/34 pages on PCA, and was not able to find an answer to this question. If its out there, my apologies, I tried.