Q1: Does A and its transpose have the same eigenspace?

  • Thread starter Thread starter Howers
  • Start date Start date
  • Tags Tags
    Transpose
Click For Summary
SUMMARY

The discussion confirms that a matrix \( A \) and its transpose \( A^T \) do not generally share the same eigenspaces, despite having identical eigenvalues due to their characteristic polynomials being the same. The distinction between right eigenvectors, which satisfy \( A\mathbf{x} = \lambda \mathbf{x} \), and left eigenvectors, which satisfy \( \mathbf{x}A = \lambda \mathbf{x} \), is emphasized. It is established that left and right eigenvectors coincide only when \( A \) is symmetric, i.e., \( A = A^T \). Furthermore, the conjecture regarding the relationship between left and right eigenvectors is clarified as being valid only under specific conditions.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors
  • Familiarity with matrix transposition
  • Knowledge of characteristic polynomials
  • Concept of symmetric matrices
NEXT STEPS
  • Study the properties of symmetric matrices and their eigenvectors
  • Learn about the relationship between left and right eigenvectors in linear algebra
  • Explore canonical forms of matrices and their implications on eigenvalues
  • Investigate methods for selecting eigenvectors of canonical lengths
USEFUL FOR

Mathematicians, linear algebra students, and anyone studying matrix theory or eigenvalue problems will benefit from this discussion.

Howers
Messages
443
Reaction score
5
So I've shown that A and A^T have the same char. polynomials => same eigenvalues, using the fact that detA = detA^T. I still can't see any way I could possibly show or disprove that the eigenspace is the same.
 
Physics news on Phys.org
How about
<br /> \left(\begin{array}{ll}<br /> 1 &amp; 1\\<br /> 0 &amp; 0<br /> \end{array}\right)\ \ <br /> \left(\begin{array}{ll}<br /> 1 &amp; 0\\<br /> 1 &amp; 0<br /> \end{array}\right)<br />
Do they have the same eigenspaces?
 
A and A^T will not have the same eigenspaces, i.e. eigenvectors, in general.

Remember that there are in fact two "eigenvectors" for every eigenvalue \lambda. The right eigenvector satisfying A\mathbf{x} = \lambda \mathbf{x} and a left eigenvector (eigenrow?) satisfying \mathbf{x}A = \lambda \mathbf{x}. In general these are not equal.

Also, I believe that the set of left eigenvectors is the inverse matrix of the set of right eigenvectors, but I am not about sure of this. If this is indeed the case then the set of left eigenvectors will "coincide" with the set of right eigenvectors only when the set of right eigenvectors is orthonormal, i.e. when A is symmetric A=A^T.

EDIT: In fact, the conjecture above is not true unless you select specially scaled sets of eigenvectors and eigenrows. Is there a way of selecting eigenvectors of canonical lengths?
 
Last edited:

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

Generalised eigenspace contains the eigenspace?
  • · Replies 2 ·
Replies
2
Views
2K
Eigenvalues of A: Same Eigenspaces for A^-1, Transpose, A^k
  • · Replies 8 ·
Replies
8
Views
3K
Show that A and its transpose have the same eigenvalues
  • · Replies 4 ·
Replies
4
Views
3K
Is Every Complex Matrix Similar to Its Transpose?
  • · Replies 2 ·
Replies
2
Views
8K
MHB Jordan Normal Form and Root Spaces
  • · Replies 11 ·
Replies
11
Views
4K
Prove or disprove: A and A transpose have the same eigenspaces
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad How Does Quantum Negativity Vary with Partial Transposes in Bipartite Systems?
  • · Replies 1 ·
Replies
1
Views
1K
Python Transpose of a non-square matrix (without using ndarray.transpose)
  • · Replies 15 ·
Replies
15
Views
5K
Undergrad Does Axler's Spectral Theorem Imply Normal Matrices?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
2K