Find Hamiltonian Value of 4x4 Matrix

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    Hamiltonian Value
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Discussion Overview

The discussion revolves around finding the Hamiltonian value of a specific 4x4 matrix, focusing on methods for determining eigenvalues. Participants explore various approaches, including perturbation theory and the Jacobi method.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks assistance in finding the value of a 4x4 matrix, requesting guidance on methods to use.
  • Another participant suggests that the matrix is Hermitian and almost block-diagonal, proposing the use of perturbation theory for estimating eigenvalues, particularly if a small parameter γ1 is considered.
  • A different participant mentions the Jacobi Method as a potential approach to solve the matrix, indicating that it may be a variation of this method.
  • One participant expresses intent to try the Hermitian method for finding the eigenvalues.
  • A later reply indicates that the matrix has been solved, though details of the solution are not provided.

Areas of Agreement / Disagreement

Participants present multiple methods for solving the problem, including the Hermitian method and the Jacobi Method, but there is no consensus on a single approach or resolution of the problem.

Contextual Notes

Participants do not clarify specific assumptions or limitations regarding the matrix or the methods discussed, leaving some aspects unresolved.

vinothr
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could you help me how to find the value of the attached 4x4 matrix.Could you give me the idea or which method i have to follow to get the value of that matrix.
 

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I'm assuming you want eigen-values. It's Hermitian and almost block-diagonal. Assuming γ1 is small, you should be able to get a good estimate, if not exact answer. A two-by-two Hermitian matrix can be diagonalized exactly by assuming that it is a rotation of a diagonal matrix by some angle theta. Solve for that angle for each of the 2x2 blocks, perform the rotations, and see what it does to your off-diagonal terms. Odds are, you'll end up with some factors proportional to γ1 scattered around. From there, your best bet is probably to go with perturbation theory to get an estimate.

Edit: Might be worth a try to use γ1 as your pivot and seeing if that leaves you with a block-diagonal matrix, but I somehow doubt it.
 
thanks for your reply.i will try Hermitian method to get eigen-value of the matrix.
 
It's Jacobi Method. Or at least a variation. If you read up on Jacobi Method, you should be able to figure out what to do with this one.
 
thanks friend.i solved the matrix.
 

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