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In this paper we address the question if, for points $P, Q \in\mathbb{P}^{2}$, $I(P)^{m} \star I(Q)^{n}=I(P \star Q)^{m+n-1}$ and we obtaindifferent results according to the number of zero coordinates in $P$ and $Q$.Successively, we use our results to define the so called Hadamard fat grids,which are the result of the Hadamard product of two sets of collinear pointswith given multiplicities. The most important invariants of Hadamard fat grids,as minimal resolution, Waldschmidt constant and resurgence, are then computed.

Hadamard products of symbolic powers and Hadamard fat grids