Invariant subspaces for tightly clustered eigenvalues of tridiagonals
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- Parlett, B.N. Bit Numer Math (1996) 36: 542. doi:10.1007/BF01731933
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The invariant subspace of a real symmetric tridiagonal matrixT associated with a tight cluster ofm eigenvalues has a special structure. This structure is revealed by the envelope of the subspace (defined in Section 3) havingm high hills separated bym − 1 low valleys.
This paper describes a long technical report that shows the existence ofm submatrices ofT each one having a single eigenvalue in the cluster interval whose normalized eigenvector has small entries in the first and last positions. These little eigenvectors determine a distinguished basis for an approximating subspace whose overlap matrix is tridiagonal and close to the identity. The only communication needed among the submatrices to make an orthogonal basis is between nearest neighbors.
A variety of examples illustrate the theory.