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

[hiroman]

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!

 

[HallsofIvy]

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 .

 

[hiroman]

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?

 

[HallsofIvy]

Yes, though in one case (stretching) the eigenvalues would be positive and in the other (contracting or compressing) they would be negative.

 

[blue_raver22]

Hello, and welcome! Eigenvalues and eigenvectors are important concepts in linear algebra and have many applications in various fields of science and engineering. In the context of a 2x2 matrix, they represent the special directions and magnitudes of transformation that the matrix performs on a vector.

To answer your first question, yes, using the tensor product is a valid way to represent two vectors on a 2x2 matrix. The resulting matrix represents the transformation that occurs when the two vectors are applied to a vector.

The eigenvalues and eigenvectors of the matrix T represent the directions and magnitudes of transformation that are preserved by the matrix. In other words, when a vector is transformed by T, the eigenvectors represent the directions that do not change and the eigenvalues represent the scaling factor of that transformation.

In terms of their relation to the original vectors v1 and v2, the eigenvectors of T are linear combinations of these vectors, and the eigenvalues represent how much of each vector is present in the eigenvector. This means that the eigenvectors of T are related to the original vectors, but they may not be exactly the same.

Repeating eigenvalues mean that there are multiple eigenvectors that represent the same direction of transformation. This can happen when the matrix has symmetry or when the transformation is along a specific axis. In terms of the original vectors, this means that there are multiple combinations of v1 and v2 that result in the same transformation.

Overall, eigenvalues and eigenvectors provide valuable information about the transformation represented by a matrix and can help us understand the underlying structure and behavior of a system. I hope this helps clarify the meaning of these concepts in relation to a 2x2 matrix.