Matrix A & A^t: Same Eigenvalues, Different Eigenvectors?

  • Thread starter gimpy
  • Start date
  • Tags
    Linear
In summary, the conversation discusses the relationship between a matrix A and its transpose A^t, specifically their eigenvalues and eigenvectors. It is shown that while the eigenvalues must be the same for both matrices, the eigenvectors may be different. An example of a 2x2 matrix where A and A^t have different eigenvectors is requested, and it is suggested that any nonsymmetric matrix would suffice.
  • #1
gimpy
28
0
I am trying to show that a matrix A and A^t have the same eigenvalues. Well I am sure its because det(a) = det(A^t), the characteristic polynomial is det(A-xI). But this question I am trying to solve also asks for an example of a 2x2 matrix A where A and A^t have different eigenvectors. I am kinda lost of this one. Wouldn't they have the same eigenvectors because they have the same eigenvalues?
 
Physics news on Phys.org
  • #2
Originally posted by gimpy
I am trying to show that a matrix A and A^t have the same eigenvalues. Well I am sure its because det(a) = det(A^t), the characteristic polynomial is det(A-xI). But this question I am trying to solve also asks for an example of a 2x2 matrix A where A and A^t have different eigenvectors. I am kinda lost of this one. Wouldn't they have the same eigenvectors because they have the same eigenvalues?

no. consider the eivenvalue equation:

[tex]A\mathbf{v}=\lambda\mathbf{v}[/tex]
then take the transpose of that equation:

[tex]\mathbf{v}^TA^T=\lambda\mathbf{v}^T[/tex]

so the eigenvector of A becomes something you might call a "left eigenvector" of A^T, but there is no reason to think that it should also be a regular eigenvector.

the eigenvalues must be the same, however, for the reason you stated.

i imagine, that to find an example of a matrix whose transpose has different eigenvectors, you should just grab any old generic nonsymmetric matrix, it will probably do.
 
Last edited:
  • #3
thanks lethe,

I feel kinda stupid that i overlooked that equation. Its really fundamental stuff[b(]
 

1. What does it mean for Matrix A and A^t to have the same eigenvalues?

Having the same eigenvalues means that the two matrices have the same set of values that satisfy the characteristic equation for each matrix. This means that the two matrices have similar properties and can potentially be used interchangeably in certain calculations.

2. Why do Matrix A and A^t have different eigenvectors despite having the same eigenvalues?

The eigenvectors of a matrix are not unique and can vary depending on the method used to calculate them. This means that even if two matrices have the same eigenvalues, the corresponding eigenvectors may not be identical.

3. Can Matrix A and A^t always be diagonalized to have the same eigenvalues and eigenvectors?

No, not all matrices can be diagonalized and have the same eigenvalues and eigenvectors. This is because diagonalization requires the matrix to have a full set of linearly independent eigenvectors, which may not always be the case.

4. How can Matrix A and A^t with the same eigenvalues but different eigenvectors be useful in real-world applications?

In certain applications, it may be more important for two matrices to have the same eigenvalues rather than identical eigenvectors. For example, in physics, two matrices with the same eigenvalues but different eigenvectors may represent two different physical systems with similar properties.

5. Is there a relationship between the eigenvectors of Matrix A and A^t?

Yes, the eigenvectors of Matrix A and A^t are related by a simple transformation. If v is an eigenvector of Matrix A with corresponding eigenvalue λ, then v is also an eigenvector of A^t with the same eigenvalue λ. Additionally, the eigenvectors of A^t can be obtained by transposing the eigenvectors of A.

Similar threads

  • Linear and Abstract Algebra
Replies
12
Views
1K
  • Linear and Abstract Algebra
Replies
24
Views
471
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
886
  • Linear and Abstract Algebra
Replies
10
Views
2K
Replies
4
Views
2K
  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
413
  • Linear and Abstract Algebra
Replies
8
Views
1K
Back
Top