Numerical ranges of 4-By-4 nilpotent matrices: Flat portions on the boundary

Erin Militzer, Linda J. Patton, Ilya M. Spitkovsky, Ming Cheng Tsai

Research output: Chapter in Book/Report/Conference proceedingChapter


In their 2008 paper Gau and Wu conjectured that the numerical range of a 4-by-4 nilpotent matrix has at most two flat portions on its boundary. We prove this conjecture, establishing along the way some additional facts of independent interest. In particular, a full description of the case in which these two portions indeed materialize and are parallel to each other is included.

Original languageEnglish (US)
Title of host publicationOperator Theory
Subtitle of host publicationAdvances and Applications
PublisherSpringer International Publishing
Number of pages31
StatePublished - 2017

Publication series

NameOperator Theory: Advances and Applications
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878


  • Nilpotent matrix
  • Numerical range

ASJC Scopus subject areas

  • Analysis


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