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We construct noncommutative U(1) gauge theory on the fuzzy sphere S^2_N as aunitary 2N x 2N matrix model. In the quantum theory the model is equivalent toa nonabelian U(N) Yang-Mills theory on a 2 dimensional lattice with 2plaquettes. This equivalence holds in the " fuzzy sphere" phase where weobserve a 3rd order phase transition between weak-coupling and strong-couplingphases of the gauge theory. In the ``matrix'' phase we have a U(N) gauge theoryon a single point.Comment: 13 pages, one grap

Quantum equivalence of noncommutative and Yang-Mills gauge theories in 2D and matrix theory