Are Eigenvalues of a Non-Hermitian Matrix Real?

  • Thread starter Thread starter andrewm
  • Start date Start date
  • Tags Tags
    Eigenvalues
Click For Summary
Eigenvalues of a non-Hermitian matrix are generally complex, and proving they are real without computation is challenging. The discussion seeks a method analogous to the Hermitian case, where real eigenvalues are guaranteed. However, no definitive criteria exist for non-Hermitian matrices that ensure real eigenvalues without calculation. Ultimately, the consensus is that computing the eigenvalues remains the most reliable approach. The topic highlights the complexities of eigenvalue properties in non-Hermitian matrices.
andrewm
Messages
49
Reaction score
0
Given a 4x4 non-Hermitian matrix, is there any method I can use to prove the eigenvalues are real, aside from actually computing them?

I'm looking for something like the converse of the statement "M is Hermitian implies M has real eigenvalues".

When can one say that the eigenvalues of a given matrix are real?
 
Physics news on Phys.org
In general, the eigenvalues of a non-Hermitian matrix can be complex. You would need to compute them.
 
OK, I'm not surprised. Thanks anyways.
 

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

Undergrad The Orthogonality of the Eigenvectors of a 2x2 Hermitian Matrix
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
2K
Undergrad Getting eigenvalues of an arbitrary matrix with programming
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Spectral theorem for Hermitian matrices-- special cases
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Types of complex matrices, why only 3?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Eigenvalues of block matrix/Related non-linear eigenvalue problem
  • · Replies 1 ·
Replies
1
Views
2K
Eigenvalues/vectors of Hermitian and corresponding unitary
  • · Replies 4 ·
Replies
4
Views
1K
MHB Show eigenvalues are real and positive (too easy?)
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad How can I test for positive semi-definiteness in matrices?
  • · Replies 1 ·
Replies
1
Views
3K