The Gau–Wu number for 4 × 4 and select arrowhead matrices

Kristin A. Camenga, Patrick X. Rault, Ilya M. Spitkovsky, Rebekah B.Johnson Yates

Research output: Contribution to journalArticlepeer-review


The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e., the maximal number k(A) of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range of A) is computed for a class of arrowhead matrices A of arbitrary size, including dichotomous ones. These results are then used to completely classify all 4×4 matrices according to the values of their Gau–Wu numbers.

Original languageEnglish (US)
Pages (from-to)192-218
Number of pages27
JournalLinear Algebra and Its Applications
StatePublished - Jul 1 2022


  • 4×4 matrices
  • Arrowhead matrix
  • Boundary generating curve
  • Field of values
  • Gau–Wu number
  • Irreducible
  • Numerical range
  • Singularity

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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