The methods of computing resolution matrices that have been illustrated here may be easily generalized to a variety of other iterative and approximate inversion methods. We have explored partial reorthogonalization methods for iterative methods and have found that using a subset of the early vectors generated in the iteration sequence is most effective at reducing unwanted occurrences of nonorthogonal vectors in the later parts of the iteration sequence. These early vectors correspond to directions that have components along the eigenvectors with the largest eigenvalues, and these are precisely the vectors we most need to exclude from the later iterations. Such recurrences may not adversely affect the inversion itself, but do make the computation of the resolution matrices (operators) much more complicated than if the orthogonalization is enforced.