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

[tex] H = \frac{1}{2}c^{\dagger}\textbf{H}c[/tex]

where [tex] c = (c_1,c_2,...c_N)^T [/tex]

The dimensions of the total Hamiltonian are 2N, because each [tex] c_i [/tex] 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 [tex] \textbf{H} [/tex] which is a tridiagonal matrix. I think my mistake is my solution attempt, I think I can't simply use like an [tex] eig(\textbf{H}) [/tex] 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!

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# I Numerical Diagonalization

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