Stewart, G. W.When the inverse power method is used to compute eigenvectors of a symmetric matrix corresponding to close eigenvalues, the computed eigenvectors may not be orthogonal. The cure for the problem is to orthogonalize the vectors using the Gram--Schmidt algorithm. In this note it is shown that the orthogonalization process does not cause the quality of the eigenvectors to deteriorate. Also cross-referenced as UMIACS-TR-99-64en-USTechnical Report