Unitary 1-matrix models are shown to be exactly equivalent to hermitian 1-matrix models coupled to 2N vectors with appropriate potentials, to all orders in the 1/N expansion. This fact allows us to use all the techniques developed and results obtained in hermitian 1-matrix models to investigate unitary as well as other 1-matrix models with the Haar measure on the unitary group. We demonstrate the use of this duality in various examples, including: (1) an explicit confirmation that the unitary matrix formulation of the N=2 pure SU(2) gauge theory correctly reproduces the genus-1 topological string amplitude (2) derivations of the special geometry relations in unitary as well as the Chern-Simons matrix models.

27 pages, 2 figures. Corrections in terminology for the mirror geometry