(no subject)

[impossiblewizardry]

I think of rotations as, in a way, not real transformations. Instead of imagining the object itself rotating, I just imagine the axes rotating. I’m just choosing to describe the same object, in a different way.

Recently I’ve had to get used to thinking of reflections in the same way too. Imagine a map of a city, and you describe points on the map as how far east of the center they are, and how far north of the center they are. Now you decide to describe them instead in terms of how far east and how far south. What you’ve done to the coordinates is a reflection. But you have not changed the map; it’s not a reflected inverted map; instead it’s only one of the basis vectors which has been reflected.

This perspective has helped me understand the terminology people use in principal component analysis. In PCA, you diagonalize the correlation matrix:

C = P D P’

The matrix P is described as the rotation matrix. But rotations all have determinant 1; might this have determinant -1 instead? At first I tried to prove somehow that the determinant must be 1 instead of -1, to explain why they call it a rotation.

But I realized, they probably make the determinant 1 by convention. You can always flip the sign of the determinant by flipping the sign of one of the columns of P. And you can always do this because P must be a basis made of eigenvectors of C, and that doesn’t change if you flip the sign of one of the basis vectors.

A symbolic way to see this is

P₂ = P₁ F

where F flips the sign of a column. And what is F? An identity matrix, but with one of the diagonal elements changed from 1 to -1. And

det P₂ = det P₁ det F

And det F is -1. So flipping the sign of a column flips the sign of the determinant.

And to see that it still works as a diagonalization:

P₂ D P₂‘ = P₁ F D F P₁ = P₁ D P₁