I am poking around at a large matrix (with dimension 2^N×2^N). The matrix is made up of ... corresponding to specific interactions

 

is it thus possible to implicitly find Eigenvalues of it, without realising it as a matrix in the first place?

If you only care about a few of the eigenvalues then you don't have to realize the operator as a 2^N x 2^N matrix in the sense of explicitly creating a dense matrix in your computer memory. You can use some ARPACK functions instead. This would not be as efficient as having an explicit formula for the eigenvalues, but it would be more efficient than constructing the matrix explicitly.

You might have better luck asking on forums other than mathoverflow.

I am poking around at a large matrix (with dimension 2^N×2^N). The matrix is made up of ... corresponding to specific interactions

is it thus possible to implicitly find Eigenvalues of it, without realising it as a matrix in the first place?

If you only care about a few of the eigenvalues then you don't have to realize the operator as a 2^N x 2^N matrix in the sense of explicitly creating a dense matrix in your computer memory. You can use some ARPACK functions instead. This would not be as efficient as having an explicit formula for the eigenvalues, but it would be more efficient than constructing the matrix explicitly.

You might have better luck asking on forums other than mathoverflow.