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Two square matrices of (arbitrary) order N are introduced. They are definedin terms of N arbitrary numbers z_{n}, and of an arbitrary additional parameter(a respectively q), and provide finite-dimensional representations of the twooperators acting on a function f(z) as follows: [f(z+a)-f(z)]/a respectively[f(qz)-f(z)]/[(q-1)z]. These representations are exact---in a sense explainedin the paper---when the function f(z) is a polynomial in z of degree less thanN. This formalism allows to transform difference equations valid in the spaceof polynomials of degree less than N into corresponding matrix-vectorequations. As an application of this technique several remarkable squarematrices of order N are identified, which feature explicitly N arbitrarynumbers z_{n}, or the N zeros of polynomials belonging to the Askey and q-Askeyschemes. Several of these findings have a Diophantine character.

Finite-dimensional representations of difference operators, and the identification of remarkable matrices