On the Gau-Wu number for some classes of matrices

Kristin A. Camenga, Patrick X. Rault, Tsvetanka Sendova, Ilya M. Spitkovsky

Research output: Contribution to journalArticlepeer-review


For a given n× n matrix A, let k(A) stand for the maximal number of orthonormal vectors xj such that the scalar products «Ax j,xj» lie on the boundary of the numerical range W(A). This number was recently introduced by Gau and Wu and we therefore call it the Gau-Wu number of the matrix A. We compute k(A) for two classes of n × n matrices A. A simple and explicit expression for k(A) for tridiagonal Toeplitz matrices A is derived. Furthermore, we prove that k(A)=2 for every pure almost normal matrix A. Note that for every matrix A we have k(A)≥2, and for normal matrices A we have k(A)=n, so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the Gau-Wu number. Finally, matrices with maximal Gau-Wu number (k(A)=n) are considered.

Original languageEnglish (US)
Pages (from-to)254-262
Number of pages9
JournalLinear Algebra and Its Applications
StatePublished - Mar 1 2014


  • Almost normal matrices
  • Numerical range
  • Toeplitz matrices

ASJC Scopus subject areas

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


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