The main objective of this paper is to develop higher order iterative methods for computing projections into invariant subspaces of a non-singular matrix. These projections can be used to determine the number of matrix eigenvalues in a given sector of the complex plane without actually computing any eigenvalue. Some of these methods are derived from applying the Newton method to simple polynomial equations with known zeros. A special emphasis is placed on computing the hermitian eigendecomposition where matrix inverse free algorithms are presented. The main results are based on computing roots of the identity matrix which commute with the given matrix. Simulations and numerical evaluation of some of the algorithms are also established.