  Calculating the observables of a Hamiltonian requires taking matrix elements
of operators in the eigenstate basis. Since eigenstates are only defined up to
arbitrary phases that depend on Hamiltonian parameters, analytical expressions
for observables are often difficult to simplify. In this work, we show how for
small Hilbert space dimension N all observables can be expressed in terms of
the Hamiltonian and its eigenvalues using the properties of the SU(N) algebra,
and we derive explicit expressions for N=2,3,4. Then we present multiple
applications specializing to the case of Bloch electrons in crystals, including
the computation of Berry curvature, quantum metric and orbital moment, as well
as a more complex observable in non-linear response, the linear photogalvanic
effect (LPGE). As a physical example we consider multiband Hamiltonians with
nodal degeneracies to show first how constraints between these observables are
relaxed when going from two to three-band models, and second how quadratic
dispersion can lead to constant LPGE at small frequencies.
