A new iterative method of finding the minimum eigenvalue of a symmetric matrix is described. This method does not utilize matrix inversions and is applicable to any matrix R for which the matrix-vector product Rx is rapidly computable. It seeks the minimum eigenvalue of R by minimizing the quadratic form X^TRx on the unit hypersphere, using a search technique derived from the conjugate gradient method. The computational complexity of each step of the algorithm depends on the speed with which Rx can be computed.