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We construct a matrix model that reproduces the topological string partitionfunction on arbitrary toric Calabi-Yau 3-folds. This demonstrates, in accordwith the BKMP "remodeling the B-model" conjecture, that Gromov-Witteninvariants of any toric Calabi-Yau 3-fold can be computed in terms of thespectral invariants of a spectral curve. Moreover, it proves that thegenerating function of Gromov-Witten invariants is a tau function for anintegrable hierarchy. In a follow-up paper, we will explicitly construct thespectral curve of our matrix model and argue that it equals the mirror curve ofthe toric Calabi-Yau manifold.

A matrix model for the topological string I: Deriving the matrix model