Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers z_n, and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two operators 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 explained in the paper---when the function f(z) is a polynomial in z of degree less than N. This formalism allows to

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... orm difference equations valid in the space of polynomials of degree less than N into corresponding matrix-vector equations. As an application of this technique several remarkable square matrices of order N are identified, which feature explicitly N arbitrary numbers z_n, or the N zeros of polynomials belonging to the Askey and q-Askey schemes. Several of these findings have a Diophantine character.