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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!

Rodrigo de Azevedo
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Belen
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    I looked a bit into it (I just googled the Russian) and I think that the term has more to do with inner product spaces than it does spectral theory. I think this because I saw a lot of relations to cosine and something that looked like a form of the Cauchy-Schwarz inequality. – Ty Jensen Dec 11 '19 at 11:46
  • Okay, thanks for the insight, I will have to read more on that because I'm only on my first semester of bacherlor's, and all those terms are greek to me. But it's helpful to at least know where to keep searching. – Belen Dec 11 '19 at 14:23
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    This sounds like the polar decomposition of a linear transformation of the plane. If its matrix is $A$, then the “principal directions” would be eigenvectors of $(A^*A)^{1/2}$. However, I’m not so sure that a general affine transformation, which might also involve a translation, can be decomposed in this way. – amd Dec 11 '19 at 22:58

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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 $T$.