We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigen\-spaces of a symmetric positive definite operator $A$ defined on a finite dimensional real Hilbert space $V$. In our applications, the dimension of $V$ is large and the cost of inverting $A$ is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning for $A$. Estimates will be provided which show that the preconditioned method converges linearly when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.