On close eigenvalues of tridiagonal matrices

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8 Scopus citations


A symmetric tridiagonal matrix with a multiple eigenvalue musthave a zerosubdiagonal element (Formula presented.) and must be a direct sum of twocomplementary blocks, both of which have the eigenvalue.Yet it is well known that a small spectral gapdoes not necessarily imply that some (Formula presented.)is small, asis demonstrated by the Wilkinson matrix.In this note, it is shown that a pair ofclose eigenvalues can only arise from twocomplementary blocks on the diagonal,in spite of the fact that the (Formula presented.)coupling thetwo blocks may not be small.In particular, some explanatory bounds are derived and aconnection tothe Lanczos algorithm is observed. The nonsymmetric problemis also included.

Original languageEnglish
Pages (from-to)507-514
Number of pages8
JournalNumerische Mathematik
Issue number4
StatePublished - Jun 1995


  • Mathematics Subject Classification (1991):65F15, 15A42

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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