How can I find the unitary matrix for diagonalizing a Hamiltonian numerically?

[DeathbyGreen]

Hi!

I'm trying to understand how to diagonalize a Hamiltonian numerically. Basically I have a problem with a Hamiltonian such as

$$H = \frac{1}{2}c^{\dagger}\textbf{H}c$$

where $$c = (c_1,c_2,...c_N)^T$$

The dimensions of the total Hamiltonian are 2N, because each $$c_i$$ is a 2 spinor. I need to numerically calculate the eigenvalues of this. My solution attempt was to simply use a QR factorization algorithm to diagonalize $$\textbf{H}$$ which is a tridiagonal matrix. I think my mistake is my solution attempt, I think I can't simply use like an $$eig(\textbf{H})$$ function. I think I need to find a unitary matrix...but I've not done this before. Is that the correct solution attempt? And if so, could someone provide an example of how to do that with the unitary matrix? Like a explicit example and solution...I would really appreciate any help!

Thank you!

 

[DeathbyGreen]

I think I've made progress in a solution but am not quite there...So my QR algorithm wasn't working because the eigenvalues come in +/- pairs on either side of the matrix diagonal. so I need to perform a Bogoliubov type transformation to find a unitary orthogonal matrix which can be used to multiply the original matrix by. My problem is that I don't know how to map my original states to the quasiparticle states. In other words:

$$(a, a^{\dagger}) = W (ua+va^{\dagger}; u^{\dagger}a-va)$$

How can I find W?