Proving kernel of matrix is isomorphic to 0 eigenvalue's eigenvectors

Click For Summary

Homework Help Overview

The discussion revolves around proving that the eigenvectors corresponding to the 0 eigenvalue of a matrix are equivalent to the kernel of that matrix. The subject area involves linear algebra, particularly concepts related to eigenvalues and eigenvectors.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definitions of the nullspace and the set of eigenvectors corresponding to the 0 eigenvalue. There is an attempt to clarify the relationship between these concepts, with some questioning the sufficiency of the original poster's proof.

Discussion Status

The discussion is ongoing, with participants providing feedback on the original poster's reasoning and seeking clearer definitions. There is no explicit consensus yet, but the dialogue indicates a productive exploration of the definitions involved.

Contextual Notes

Participants are examining the definitions of the kernel of a matrix and the eigenvectors associated with the 0 eigenvalue, suggesting that assumptions about their equivalence may need further scrutiny.

Coolphreak
Messages
44
Reaction score
0

Homework Statement


I want to prove that the eigenvectors corresponding to the 0 eigenvalue of hte matrix is the same thing as the kernel of the matrix.


Homework Equations


A = matrix.
L = lambda (eigenvalues)

Ax=Lx


The Attempt at a Solution



Ax = 0 is the nullspace.

Ax = Lx
Lx = 0.
L= 0.
the eigenvectors corresponding to the 0 eigenvalue are the same as the nullspace.

Is this a sufficient enough proof?
 
Physics news on Phys.org
No, it's not. Maybe you have the right idea, but what you've written down doesn't make a lot of sense.

The nullspace is {x : Ax = 0}. Can you write down what the set of eigenvectors corresponding to zero is?
 
Isn't the the set of eigenvectors which correspond to the 0 eigenvalue?
 
What is the definition of the kernel of a matrix? What is the definition of the set of eigenvectors of a matrix with eigenvalue zero? Aren't they trivially the same?
 

Similar threads

Replies
5
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
Replies
2
Views
1K
Replies
8
Views
3K
  • · Replies 7 ·
Replies
7
Views
1K
  • · Replies 5 ·
Replies
5
Views
3K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 4 ·
Replies
4
Views
3K
  • · Replies 19 ·
Replies
19
Views
4K
Replies
2
Views
2K