The Construction of Orthogonal Eigenvectors for Tight Clusters by Use of Submatrices,

Abstract
The goal is to compute eigenvectors of a symmetric tridiagonal matrix T that are orthogonal to working accuracy. Consider a cluster of m very close eigenvalues that are reasonably well separated from the remaining spectrum. We show here that there are m principal submatrices of T such that only the nearest neighbors overlap and each submatrix has a simple, isolated eigenvalue in the convex hull of the cluster with eigenvector having small entries in the first and last positions. This eigenvector is padded with zero entries, above and below, to make it conform to T. The set of vectors, one from each submatrix, forms a good basis for the invariant subspace. Each basis vector may be modified, if necessary, by its nearest neighbors to produce an orthonormal basis. The only communication that may be needed, in such situations, is between nearest neighbors. We give a good bound on the dot product of nearest neighbors. A variety of examples illustrate the theory.

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