Alternating minimization (AltMin) algorithms are proposed for hybrid precoding in millimeter wave (mmWave) MIMO systems. The paper addresses the design of hybrid precoders for two structures: fully-connected and partially-connected. For the fully-connected structure, an AltMin algorithm based on manifold optimization (MO-AltMin) is introduced, which directly solves the hybrid precoder design problem under unit modulus constraints. This algorithm does not require a pre-determined candidate set for the analog precoder and is the first to directly solve the problem under these constraints. A low-complexity counterpart, PE-AltMin, is also proposed by enforcing an orthogonal constraint on the digital precoder. For the partially-connected structure, a semidefinite relaxation based AltMin (SDR-AltMin) algorithm is developed, which effectively designs the hybrid precoders by solving both subproblems in each iteration. The proposed algorithms are extended to broadband OFDM systems. Simulation results show that the MO-AltMin algorithm efficiently identifies a near-optimal solution, while the PE-AltMin algorithm outperforms existing algorithms in terms of performance. The algorithms are compared to reveal design insights, showing that the fully-connected structure can approach the performance of fully digital precoders when the number of RF chains is comparable to the number of data streams. The partially-connected structure provides significant gains over analog beamforming and higher energy efficiency. The results establish the effectiveness of alternating minimization as a key design methodology for hybrid precoding in mmWave MIMO systems.Alternating minimization (AltMin) algorithms are proposed for hybrid precoding in millimeter wave (mmWave) MIMO systems. The paper addresses the design of hybrid precoders for two structures: fully-connected and partially-connected. For the fully-connected structure, an AltMin algorithm based on manifold optimization (MO-AltMin) is introduced, which directly solves the hybrid precoder design problem under unit modulus constraints. This algorithm does not require a pre-determined candidate set for the analog precoder and is the first to directly solve the problem under these constraints. A low-complexity counterpart, PE-AltMin, is also proposed by enforcing an orthogonal constraint on the digital precoder. For the partially-connected structure, a semidefinite relaxation based AltMin (SDR-AltMin) algorithm is developed, which effectively designs the hybrid precoders by solving both subproblems in each iteration. The proposed algorithms are extended to broadband OFDM systems. Simulation results show that the MO-AltMin algorithm efficiently identifies a near-optimal solution, while the PE-AltMin algorithm outperforms existing algorithms in terms of performance. The algorithms are compared to reveal design insights, showing that the fully-connected structure can approach the performance of fully digital precoders when the number of RF chains is comparable to the number of data streams. The partially-connected structure provides significant gains over analog beamforming and higher energy efficiency. The results establish the effectiveness of alternating minimization as a key design methodology for hybrid precoding in mmWave MIMO systems.