Geometric Parter–Wiener, etc. theory

Charles R. Johnson, Carlos M. Saiago

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


For real symmetric, or complex Hermitian, matrices whose graph is a tree, there is a well-developed (via several papers) theory about the possible multiplicities of the eigenvalues. It includes a theory of vertices whose removal increases a multiplicity (the “Parter–Wiener, etc. theory” and the “downer branch mechanism”), how to determine maximum multiplicity, lower bounds for the minimum number of distinct eigenvalues, etc. Remarkably, a great deal of this theory may be generalized to geometric multiplicities of general matrices over a field (with very different proofs). We show here what parts of this theory generalize, and, in the process, review the theory.

Original languageEnglish
Pages (from-to)332-347
Number of pages16
JournalLinear Algebra and its Applications
Publication statusPublished - 15 Jan 2018


  • Eigenvalue
  • General matrix
  • Geometric multiplicity
  • Maximum multiplicity
  • Parter vertex
  • Tree


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