Find Hamiltonian Value of 4x4 Matrix

[vinothr]

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.

 

[K^2]

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.

 

[vinothr]

thanks for your reply.i will try Hermitian method to get eigen-value of the matrix.

 

[K^2]

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.

 

[vinothr]

thanks friend.i solved the matrix.