How to best define a “contrast” in a Principal Component Analysis (PCA)?

I have been studying how to interpret principal components.

I recently came across an example of a particular eigenvector:

$$e_j^T = \left[ \frac{\sqrt{2} }{2}, \frac{-\sqrt{2} }{2}, 0, \dots,0 \right]$$

I am under the impression that, often in social sciences, "this is called a contrast."

From what I can see, we have the difference between $X_1$ and $X_2$ (variables in our original dataset) playing a role in the $j$th eigenvector.

What can this say about the relationship between $X_1$ and $X_2$, with respect to this eigenvector? There is clearly something going on with the equal, yet opposite, contribution of these independent variables, to our (possibly first) principal component. I guess the answer to this is the answer to the question, "what is a contrast?"

Strangely enough, googling hasn't coming up with much of anything aside from dense discussion such as found on the Wikipedia page.

• Some corrections are needed to stave of possible confusion: first, $e_j^\prime$ is an eigenvector, not an eigenvalue. Second, it does not assert any relationship at all between $X_1$ and $X_2$: it is literally just a linear combination of the two. FWIW, googling "linear regression test contrast" turns up the Wikipedia article on the subject right away. – whuber Mar 29 '14 at 20:54
• argh, thanks, I've fixed my typos and generally cleaned up the question. – Matt O'Brien Mar 29 '14 at 21:34
• I'm confused...I was taught it's dangerous to interpret PCA components because they are not identifiable, only the subspace they span is identifiable. – anonymous Jun 5 '15 at 18:49
• @user43228 That's right. I think it's understood that $e_j^T$ is being considered only up to a nonzero multiple. The usual way to express it as a contrast would be as the vector $(1,-1,0,\ldots,0)$ or $(-1,1,0,\ldots,0)$. That is why Matt (the OP) is using terms like "equal, yet opposite contribution" without focusing on the magnitudes of the components. – whuber Jun 5 '15 at 18:56
• @aabeshou On the contrary, one can interpret individual PCA components (and this is often done). Nevertheless, I agree with your question and answer. – whuber Jun 5 '15 at 20:48