BIT Numerical Mathematics

, Volume 36, Issue 3, pp 542–562

Invariant subspaces for tightly clustered eigenvalues of tridiagonals

  • B. N. Parlett

DOI: 10.1007/BF01731933

Cite this article as:
Parlett, B.N. Bit Numer Math (1996) 36: 542. doi:10.1007/BF01731933


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.

Copyright information

© BIT Foundation 1996

Authors and Affiliations

  • B. N. Parlett
    • 1
  1. 1.Mathematics Department and Computer Science Division, EECS DepartmentUniversity of CaliforniaBerkeleyUSA