I How do i find the eigenvalues of this tough Hamiltonian?

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I have this Hamiltonian --> (http://imgur.com/a/lpxCz)

Where each G is a matrix.

I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain explicitly on the computer it would be a 6D tensor right? So how do I change it into a matrix so I can find the eigenvalues?

One thing I've been told it that I could build a matrix with the i' indices going down and the i indices going across. Neglecting the sum over the alphas, if I did this would the first row of my matrix be (G1,1 G1,1G1,1,G1,1 G1,1 G1,2,G1,1 G1,1 G1,3,...,G1,1 G1,1 G1,10,G1,1 G1,2 G1,1,...,G1,1 G1,2 G1,10,...,G1,1 G1,10 G1,10,G1,2 G1,1G1,1,...G1,10 G1,10 G1,10)
where each element is the product of the elements of the G matrices. I don't see you can take the product of the elements of the G matrices if they aren't defined in the same space. Would constructing a matrix like this really give the eigenvalues of H?
 
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 

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