# Comparing original variables with characteristic values of diagonalized variance-covariance matrix

If I have a reference data set comprising repeated measurements of 3 variables of a system in state $A$. Given new observations of these variables for a different system I would like to classify individual observations as being in state $A$ or not.

My initial inclination would be to compare the new value of each variable to the distribution in the reference state. However, a manuscript I am reading for a similar type of analysis suggests to instead construct a $3 \times 3$ covariance matrix based on the reference state, and diagonalize this covariance matrix. I am advised that I can compare each new set of measurements of the 3 variables against this diagonalized matrix. My understanding is that to follow this method, I need to examine the relative deviation of the new variables ($x_i$) from the eigenvalues ($\lambda_i$) of the new matrix and determine if this deviation is within a certain tolerance.

$\sum_{i=1}^3 \left(x_i - \sqrt{\lambda_i}\right)^2$

It seems that in this case I am comparing a set of new measurements $x_i$ against the loadings of a set of transformed variables (according to my interpretation of PCA).

Is this a common method of classification, and is there a term for this type of analysis? I was not aware that the singular values (square root of eigenvalues) would be directly comparable to the original variables.