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

Click For Summary
SUMMARY

This discussion focuses on the interpretation of eigenvalues and eigenvectors in the context of a 2x2 matrix constructed using the tensor product of two 2-dimensional vectors, "v1" and "v2". The matrix "T" is defined as T = v1 ⊗ v2. Participants clarify that eigenvalues indicate the magnitude of transformation along the eigenvectors, which represent principal directions in the transformation. The discussion also addresses the significance of repeated eigenvalues, indicating that any eigenvector can be associated with the matrix, and emphasizes the geometric interpretation of these concepts using the analogy of warping a rubber sheet.

PREREQUISITES
  • Understanding of Linear Algebra concepts, specifically eigenvalues and eigenvectors.
  • Familiarity with tensor products, particularly in the context of 2x2 matrices.
  • Basic knowledge of geometric interpretations of linear transformations.
  • Ability to compute eigenvalues and eigenvectors for matrices.
NEXT STEPS
  • Study the properties of tensor products in linear algebra.
  • Explore the geometric interpretation of eigenvalues and eigenvectors in greater detail.
  • Learn about the implications of repeated eigenvalues in matrix theory.
  • Investigate applications of eigenvalues and eigenvectors in various fields, such as physics and engineering.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of linear transformations, eigenvalues, and eigenvectors, particularly in the context of 2x2 matrices and tensor products.

hiroman
Messages
6
Reaction score
0
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!
 
Physics news on Phys.org
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.
 

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
View full post »

Similar threads

Undergrad Is a 1x1 Matrix Considered a Scalar in Mathematics?
  • · Replies 12 ·
Replies
12
Views
3K
Eigenvector of Pauli Matrix (z-component of Pauli matrix)
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
2K
Undergrad Question Regarding Definition of Tensor Algebra
  • · Replies 10 ·
Replies
10
Views
3K
Vector iteration/Power method for eigen values
  • · Replies 6 ·
Replies
6
Views
6K
What is the Correct Way to Find Eigenvalues and Eigenvectors of a Matrix?
  • · Replies 1 ·
Replies
1
Views
2K
High School Array Representation Of A General Tensor Question
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Eigenvectors and matrix inner product
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Orthonormal basis expression for ordinary contraction of a tensor
  • · Replies 2 ·
Replies
2
Views
2K