The properties and structure of the Hamiltonian matrix of a realistic many-body quantum chaotic system (the rare-earth atom of Ce) are analyzed and compared with those assumed in band random matrix theories. The sparsity of the matrix and the behavior of the mean squared matrix elements 〈${\mathit{H}}_{\mathit{i}\mathit{j}}^2$${\mathrm{〉}}_{\mathrm{\Vert }\mathit{i}\mathrm{-}\mathit{j}\mathrm{\Vert }=\mathrm{\Delta }}$ as function of the distance ‖i-j‖ from the diagonal are studied. Fitting 〈${\mathit{H}}_{\mathit{i}\mathit{j}}^2$${\mathrm{〉}}_{\mathrm{\Vert }\mathit{i}\mathrm{-}\mathit{j}\mathrm{\Vert }=\mathrm{\Delta }}$ with the exponent ${\mathit{H}}_0^2$exp(-Δ/b) yields the bandwidths of b=54 and 79 for the matrices of ${\mathit{J}}^{\mathrm{\pi }}$=$4^{+}$, and $4^{\mathrm{-}}$ states, respectively.

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