A simple characterization of uniformly minimum variance unbiased estimators (UMVUEs) is provided (in the case when the sample space is finite) in terms of a linear independence condition on the likelihood functions corresponding to the possible samples. The crucial observation in the proof is that, if a UMVUE exists, then, after an appropriate cleaning of the parameter space, the nonzero likelihood functions are eigenvectors of an "artificial" matrix of Lagrange multipliers, and the values of the UMVUE are eigenvalues of that matrix. The characterization is then extended to best unbiased estimators with respect to arbitrary convex loss functions.