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The paper characterizes matrices 
which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a 0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij } satisfying p 00 + p 01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.
