Meaning of eigenvectors and values of a 2x2 matrix (2nd order tensor)

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Discussion Overview

The discussion revolves around the meaning and implications of eigenvalues and eigenvectors in the context of a 2x2 matrix, specifically one constructed from the tensor product of two vectors. Participants explore the relationship between the matrix and its eigenvalues/vectors, as well as the significance of repeated eigenvalues.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • A new user questions whether using the tensor product to create a 2x2 matrix from two vectors is appropriate and seeks clarification on the meaning of eigenvalues and eigenvectors in this context.
  • Some participants propose that eigenvectors represent principal directions in the transformation described by the matrix, with eigenvalues indicating the extent of movement along these directions.
  • A participant suggests that eigenvectors may remain the same under different orientations, raising the question of how stretching or contracting affects their interpretation.
  • Another participant affirms that the nature of eigenvalues (positive for stretching, negative for contracting) influences the interpretation of the eigenvectors.

Areas of Agreement / Disagreement

Participants generally agree on the conceptual framework of eigenvalues and eigenvectors but express differing views on the implications of orientation and the meaning of repeated eigenvalues. The discussion remains unresolved regarding the deeper significance of these concepts in relation to the original vectors.

Contextual Notes

Limitations include potential misunderstandings of the tensor product's implications, the dependence on specific definitions of eigenvalues and eigenvectors, and the unresolved nature of how these concepts relate to the original vectors used to construct the matrix.

hiroman
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Hi! I am a new user who is not an expert with Linear Algebra at all.

I have some questions about eigen values/vectors and their meaning with relation to a 2x2 matrix, or tensor, which was obtained by the tensor product of 2 vectors.

First, I have two 2-dimensional 2x1 vectors "v1" and "v2" on one point from which I wish to construct a 2x2 matrix "T" using tensor product, ie T=v1 (circle x) v2.

Then, I compute the eigen values and eigen vectors of the matrix (tensor) T.

Questions:

Is using tensor product the correct way to represent the vectors v1 and v2 on a 2x2 matrix T?

What's the meaning of the eigen values and eigen vectors of T? What is their relation with the original vectos v1 and v2? Also, most importantly, what is the meaning of having eigen values that are repeated?

I have read that if the eigen values of T are repeated, then that means that any eigen vector is associated with T, but still cannot figure out its underlying meaning with respect to the original vectors that constructed T.

Thanks!
 
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Geometrically, we can think of a two by two matrix. A, as "warping" points in a plane. If u and v are eigenvectors of A then they point in the "principal directions" of A. Think of four people holding a rubber sheet and pulling on it. Points on the line connecting two diagonally opposite people are just moved along that line. Those lines are in the direction of the eigenvectors and and the eigenvalues tell how far they are moved. Other points are moved part toward one person and partly toward another .
 
Thanks for the illustration on eigenvalues and eigenvectors. Then, is it correct to consider that eigenvectors of a matrix are the same if the orientation is different? Per the example, the eigenvectors would be the same if the people are stretching or contracting the rubber sheet?
 
Yes, though in one case (stretching) the eigenvalues would be positive and in the other (contracting or compressing) they would be negative.
 

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