The Parter-Wiener theorem: Refinement and generalization

Charles R. Johnson, António Leal-Duarte, Carlos Manuel Saiago

Research output: Contribution to journalArticlepeer-review

90 Citations (Scopus)
73 Downloads (Pure)


An important theorem about the existence of principal submatrices of a Hermitian matrixwhose graph is a tree, in which the multiplicity of an eigenvalue increases, was largely developed in separate papers by Parter and Wiener. Here, the prior work is fully stated, then generalized with a self-contained proof. The more complete result is then used to better understand the eigenvalue possibilities of reducible principal submatrices of Hermitian tridiagonal matrices. Sets of vertices, for which the multiplicity increases, are also studied.
Original languageEnglish
Pages (from-to)352-361
Number of pages10
JournalSiam Journal On Matrix Analysis And Applications
Issue number2
Publication statusPublished - 2004


  • Eigenvalues
  • Hermitian matrix
  • Multiplicity
  • Parter vertices
  • Tree
  • Vertex degrees


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