Explicit matrix representation for the Hamiltonian of the one dimensional \mbox{spin~$1\mathrm{/}2$} Ising model in mutually orthogonal external magnetic fields

We derive an explicit matrix representation for the Hamiltonian of the Ising model in mutually orthogonal external magnetic fields, using as basis the eigenstates of a system of non-interacting \mbox{spin~$1\mathrm{/}2$} particles in external magnetic fields. We subsequently apply our results to obtain an analytical expression for the ground state energy per spin, to the fourth order in the exchange integral, for the Ising model in perpendicular external fields.