# How to interpret the loading values of a pca?

Imagine I've the following matrix, which gives the grades of students in the subjects German, Philosophy, Math and Physics:

ger = c(2,4,1,3,2,4,4,1,2,3)
phi = c(3,4,1,2,2,3,3,2,2,2)
mat = c(1,3,2,4,1,2,2,4,3,1)
phy = c(2,2,2,5,2,2,3,4,3,3)
A = cbind(ger,phi,mat,phy)

I combine everything to a matrix and scale the data:

As = scale(A)

Now, I perform a summary on the PCA:

Which returns the following output:

Importance of components:
Comp.1    Comp.2     Comp.3     Comp.4
Standard deviation     1.3257523 1.1657791 0.59600603 0.35793402
Proportion of Variance 0.4882275 0.3775114 0.09867311 0.03558799
Cumulative Proportion  0.4882275 0.8657389 0.96441201 1.00000000

Comp.1 Comp.2 Comp.3 Comp.4
ger  0.496 -0.502  0.519  0.482
phi  0.548 -0.443 -0.423 -0.570
mat  -0.430 -0.572 -0.546  0.435
phy  -0.518 -0.474  0.503 -0.503

I have a few hints for the first component (based on the loadings):

• There is a high positive correlation between german and philosophy and there is also a high positive correlation between math and physics.
• Who is good in language (german and philosophy) is often worse in MINT (math and physics) and the other way around.

And an idea about the second one, which I cannot interpret:

• It's a weighted arithmetic mean over all four variables.

But I have no idea how to interpret the Comp. 2, Comp. 3 and Comp. 4 based on the loadings. Especially because all values of Comp. 2 are all negative, or have the same orientation. Can someone help me? Thanks in advance!

For each student, we can define a point in a four-dimensional space (specifically, in $\mathbb{R}^4$) which represents their grades (after centering and normalization). Or to put it another way, you can imagine the set of all students' grades as a scatterplot in four dimensions, with four perpendicular axes. We can orient these axes in various directions (just as we can in two or three dimensions). The most obvious choice is to have one axis for each subject, so the axis which is collinear with the unit vector pointing from the origin to the point $(1,0,0,0)$ represents their grade in German, and likewise the axis which is collinear with the vector $(0,1,0,0)$ represents their grade in Philosophy, the axis which is collinear with the vector $(0,0,1,0)$ represents their grade in Math, and the axis which is collinear with the vector $(0,0,0,1)$ represents their grade in Physics.