  For a family of polynomials in two variables, orthogonal with respect to a
weight function, we prove under some conditions, equivalence between: the
Matrix Pearson equation of the weight, the second order linear partial
differential equation, the orthogonality of the gradients, the Matrix Rodrigues
formula involving tensor product of Matrices, and the so-called first structure
relation. We then propose a definition of classical orthogonal polynomials in
two variables.
