Finite commuting normal operators have joint spectral projections ($T_i={\sum }_j{\lambda }_{ij}{P}_j,P_j^{\ast }=P_j,P_j{P}_k={\delta }_{jk}{P}_j$)

A finite family of pairwise commuting normal operators on a finite-dimensional complex inner product space, commuting also with each other's adjoints, generates a commutative unital C-star algebra. Its finite maximal ideal space identifies the algebra with functions on finitely many points. The characteristic functions of those points pull back under the Gelfand transform to mutually orthogonal self-adjoint linear projections summing to the identity. Their ranges are the simultaneous eigenspaces, giving simultaneous diagonalization without choosing separate incompatible eigenbases.