Diagonalize a non-hermitian matrix

  • Context: Graduate 
  • Thread starter Thread starter njuclean
  • Start date Start date
  • Tags Tags
    Matrix Non-hermitian
Click For Summary

Discussion Overview

The discussion revolves around the diagonalization of non-Hermitian matrices, particularly in the context of transformation optics and the properties of eigenvectors and eigenvalues. Participants explore the implications of using unitary matrices for diagonalization and the conditions under which left and right eigenvectors form bases.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions their understanding of how the unitary matrix S, defined as the sum of outer products of right and left eigenvectors, can diagonalize the non-Hermitian transfer matrix T.
  • Another participant argues that S cannot be the unit matrix if the left and right eigenvectors do not coincide, suggesting that S's operation transforms left eigenvectors into corresponding right eigenvectors.
  • A participant notes that left and right eigenvectors do not form an orthogonal basis for all non-Hermitian operators, but only for a subclass known as "normal" operators, which can be expressed as the sum of Hermitian operators.
  • One participant provides an example of a simple two-dimensional non-Hermitian matrix and calculates its eigenvalues and eigenvectors, concluding that the sum of the outer products yields the identity matrix.
  • Another participant expresses a desire to improve their understanding of linear algebra in light of the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the properties of left and right eigenvectors in relation to non-Hermitian matrices, indicating that there is no consensus on the general behavior of these eigenvectors.

Contextual Notes

Participants acknowledge limitations in their understanding of linear algebra, which may affect their interpretations of the properties of non-Hermitian matrices and their eigenvectors.

njuclean
Messages
3
Reaction score
0
I'm learning "the transformation optics" and the first document about this method is "Photonic band structures" ( Pendry, J. B. 1993). In this document, the transfer matrix T is non-hermitian, Ri and Li are the right and left eigenvectors respectively.

Pendry defined a unitary matrix S=\sumRiLi, here RiLi is the outer product between the right eigenvector and the left eigenvector. Then T can be diagonalized through STS-1.

I remember that the sum of the outer product between the right eigenvector and the left eigenvector is the unit matrix, so i cannot understand how can the unitary matrix S diagonalize T. Who can tell me what's wrong in my understanding?
 
Last edited:
Physics news on Phys.org
In general, S won't be the unit matrix. Consider it's operation on one of the left eigenvectors L_i. It will get transformed in the corresponding right eigenvector R_i. If the left and right eigenvectors do not conincide, S cannot be the unit matrix.
 
DrDu said:
In general, S won't be the unit matrix. Consider it's operation on one of the left eigenvectors L_i. It will get transformed in the corresponding right eigenvector R_i. If the left and right eigenvectors do not conincide, S cannot be the unit matrix.

Thanks for your reply.

Are the right vectors of a non-hermitian matrix in general not the orthogonal and complete bases? I'm lack of the knowledge of linear algebra, so my question maybe too simple.
 
The left or right eigenvectors form an orthogonal basis not for all possible non-Hermitean operators but only for a sub-class, the so-called "normal" operators. All normal operators can be expressed in the form A+iB where A and B are Hermitean operators.
Maybe you should try to work out the eigenvalues and eigenvectors for some very simple two dimensional matrices e.g. one with i and -i on the diagonal and zero on the outer diagonal.
 
I'm glad to see your reply again.

The eigenvalues of the two dimensional matrix (a11=i, a12=0, a21=0, a22=-i) are \lambda1=i and \lambda2=-i. For \lambda1, R1=(1 0)T and L1=(1 0). For \lambda2, R2=(0 1)T and L2=(0 1). Thus \sumRjLj=R1L1+R2L2=I.

I think it's necessary for me to read some books about the linear algebra. Anyway, thanks again for your help.
 
Last edited:

Similar threads

Graduate Can a Non-Diagonal Hermitian Matrix be Diagonalized Using Unitary Matrix?
  • · Replies 9 ·
Replies
9
Views
3K
Graduate Diagonalization of 2x2 Hermitian matrices using Wigner D-Matrix
  • · Replies 4 ·
Replies
4
Views
5K
How can a diagonalising matrix be unitary?
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Matrix construction for spinors
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Spectral theorem for Hermitian matrices-- special cases
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Understanding how quantum annealing solves QUBO problems
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
4K
Hermitian Matrix and Commutation relations
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Hermitian operators, matrices and basis
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Eigenproblem for non-normal matrix
  • · Replies 5 ·
Replies
5
Views
3K