The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: $$H_i=\sum_{q=1}^\infty \sigma_{q,i} \; \lambda_q$$ while $q$ and $i$ positive integers.

What we find is that $\sigma_{q,i}$ is an infinite matrix, and we have all elements of this infinite matrix from observations. $\sigma_{q,i}$ contains only positive integers (and zero), it is a lower triangular matrix with all diagonal elements equal zero. The matrix is not random.

I wonder if you can help me to understand where and how such matrices are dealt with in math/physics? Is there a specific name/notation for them? Is there a certain intuition behind of such Hamiltonian and such type of matrix? I want to analyse $\sigma_{q,i}$ further from different point of view.

Thanks

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A lower triangular matrix as a Hamiltonian, really? Hasnâ€™t the Hamiltonian be Hermitian? – Incnis Mrsi Apr 22 '15 at 12:06