Embarassing question about eigenvectors

In summary, the conversation discusses the concept of spinors, which are spin 1/2 state vectors that exist in 2 dimensional complex space. The problem at hand is finding the eigenvalues of a non-square matrix, which is causing confusion as eigenvectors only make sense for square matrices. The solution is suggested to be using the Pauli spin-matrices.
  • #1
Ed Quanta
297
0
Ok, so let us suppose we have a spinor which is a spin 1/2 state vector

(a)
(b)

Now spinors exist in 2 dimensional complex space. How do I find the eigenvalues which correspond to the above eigenvector



I am confused because we are dealing with eigenvalues for a matrix which is not a square matrix. I know for a square matrix we just find the eigenvalues such that the determinant of the matrix becomes zero. I am not sure how to deal with determinants of non-square matrices however. Helo anybody?
 
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  • #2
Eigenvectors only make sense for square matrices. Think about it. Suppose you have a mxn matrix and a nx1 vector. Multiplication will give a mx1 vector. In any case not a scalar multiple of itself.

The eigenvalues are determined by the matrix. not by the eigenvector itself. Si I can't answer the question:'How do I find the eigenvalues which correspond to the above eigenvector ' if I don't know the matrix.

EDIT: Whoops. This is physics ofcourse. The matrices you need are probably the Pauli spin-matrices. Use those.
 
Last edited:
  • #3
See the other post on this subject...
 

1. What are eigenvectors and why are they important?

Eigenvectors are a type of vector that represent the direction of the linear transformation of a matrix. They are important because they help us understand the behavior of a matrix and are used in many fields such as physics, engineering, and computer science.

2. How do I find the eigenvectors of a matrix?

To find the eigenvectors of a matrix, you first need to find the eigenvalues. This can be done by solving the characteristic equation of the matrix. Once you have the eigenvalues, you can plug them back into the matrix to find the corresponding eigenvectors.

3. Can a matrix have more than one eigenvector?

Yes, a matrix can have multiple eigenvectors. In fact, the number of eigenvectors is equal to the number of eigenvalues, and each eigenvalue can have multiple corresponding eigenvectors.

4. What is the significance of the magnitude of an eigenvector?

The magnitude of an eigenvector does not have any special significance on its own. However, the direction of the eigenvector is important as it represents the direction of the linear transformation of the matrix.

5. Do all matrices have eigenvectors?

No, not all matrices have eigenvectors. For a matrix to have eigenvectors, it must be square (number of rows = number of columns) and have distinct eigenvalues. Matrices that do not meet these criteria do not have eigenvectors.

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