The authors introduce an alternative method for solving the extreme eigenvalue problem for DOA estimation. The proposed technique utilizes the conjugate gradient method (CGM) for iteratively and efficiently finding one of the noise eigenvectors which corresponds to the smallest eigenvalue of the autocovariance matrix which is full complex-valued semidefinite Hermitian. When the proposed minimum eigenvalue searching (MES) method is utilized, only one noise eigenvector which corresponds to the smallest eigenvalue is computed. In addition to circumventing the need for a separate routine, the MES method can estimate the DOA of fully coherent signals without including the detection procedure under the assumption that the number of unknowns involved in the procedure is larger than the actual number of signals. The authors also provide a suggestion for removing the pseudopeaks appearing in the spatial spectrum due to the large number of antenna elements.