http://www.amazon.com/review/R16RJ2PT63DZ3Q/ref=cm_cr_rev_de...
It seems like PCA would already be a method that would only mean something if it was applied to comparable dimensions. What would transformed, dimensioned variables mean anyway? Chart A=mass - 3charge by B = mass + 2charge. What could a correlation mean.
PCA is a form of (or at least related to) correlation. With standardization the resulting transformation hihlights variables in the original data that are most highly correlated. Without standardization you're visualizing covariation. Unlike correlation, covariation is influenced by the magnitude of the variables.
By standardizing, you control for differences in the magnitude of the variables, and focus on their inherent variation instead.
However, let's set that aside. I apologize for being a bit obfuscatory. My point is: If this is the case, then the explanation in the OP is totally misleading, because your data shouldn't look like an ellipsoid, but rather a circle. PCA should only be used in situations where there is a reason to believe there is a mechanistically justifiable "hidden value" that underlies otherwise uncontrolled "independent variables", thus making a dimensional reduction reasonable.
This is not at all the situation that the OP goes over in the first part of the post.
This is easily grasped with a 2d example, despite the fact that PCA makes no sense with only two variables.