On computing eigenvectors of symmetric tridiagonal matrices

The computation of the eigenvalue decomposition of symmetric matrices is one of the most investigated problems in numerical linear algebra. For a matrix of moderate size, the customary procedure is to reduce it to a symmetric tridiagonal one by means of an orthogonal similarity transformation and then compute the eigendecomposition of the tridiagonal matrix. Recently, Malyshev and Dhillon have proposed an algorithm for deflating the tridiagonal matrix, once an eigenvalue has been computed. Starting from the aforementioned algorithm, in this manuscript we develop a procedure for computing an eigenvector of a symmetric tridiagonal matrix, once its associate eigenvalue is known. We illustrate the behavior of the proposed method with a number of numerical examples.