= Finite commuting normal operators have joint spectral projections
{title2=$T_i=\sum_j\lambda_{ij}P_j,\quad P_j^*=P_j,\quad P_jP_k=\delta_{jk}P_j$}
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.
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