How do i find the eigenvalues of this tough Hamiltonian?

[baouba]

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 (G^1,1 G^1,1G^1,1,G^1,1 G^1,1 G^1,2,G^1,1 G^1,1 G^1,3,...,G^1,1 G^1,1 G^1,10,G^1,1 G^1,2 G^1,1,...,G^1,1 G^1,2 G^1,10,...,G^1,1 G^1,10 G^1,10,G^1,2 G^1,1G^1,1,...G^1,10 G^1,10 G^1,10)
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?

 

[azntoon]

The other thing I'm thinking might work is to use some sort of tensor contraction. Is there a way to do this? I'm not sure how to approach this problem, any help would be greatly appreciated!