Expanding Hermitian operators in a basis of projectors on coherent spin states

The expectation values of a Hermitian operator A in (2s+1)2 specific coherent states of a spin are known to determine the operator unambiguously. As shown here, (almost) any other set of (2s+1)2 coherent state projectors also provide a basis for self-adjoint operators. This is proved by considering the determinant of the Gram matrix associated with the coherent state projectors as a Hamiltonian of a fictitious classical spin system. The result guarantees that (almost) any experimentally desirable choice of directions is appropriate for reconstructing the state of a quantum spin by means of a Stern-Gerlach apparatus