You can define a given an affine transformation $f$ as a combination of an orthogonal transformation $g$ (distance between points are conserved) and two compressions $h_1,h_2$ to two mutually perpendicular straight lines with a certain coefficient of compression $\lambda$.
In russian textbooks they define 'main directions' (главные направления) as the direction vectors of these two lines.
I wanted to read more on this topic in English because I don't understand everything in Russian, but I couldn't find it. Maybe I'm not translating the terms correctly.
My question is, does this have anything to do with eigenvectors and eigenvalues? Are coefficients of compression $\lambda$ the same thing as the eigenvalues?
Thank you so much for your help!
From Vaisman, I. (1997). Analytical Geometry:
A vector $\vec v \neq 0$ defines a principal direction of a conic (quadric) $\Gamma$ iff there exists a number $s$ such that $T\vec v = s\vec v$. For the same vectors $\vec v$, another name is provided by linear algebra: eigenvectors. The value of $s$ to which eigenvector $\vec v$ corresponds is called the corresponding eingenvalue of the operator $T$.