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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

^{1,1}G

^{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

^{1,1}G

^{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?