This paper concerns with the solution of a special eigenvalue problem for a large sparse symmetric matrix by a fast convergent minimization method. A theoretical analysis of the method is developed; it is proved that is convergent with a convergence rate of fourth order. This minimization method requires to solve a sequence of equality-constrained least squares problems that become increasingly ill-conditioned, as the solution of eigenvalue problem is approached, A particular attention has been addressed to this question of ill-conditioning for the practical application of the method. Computational experiments carried out on Cray C90 show the behaviour of this minimization method as accelerating technique of the inverse iteration method. Also a comparison with the scaled Newton method has been done.