### Abstract

For a given n× n matrix A, let k(A) stand for the maximal number of orthonormal vectors x^{j} such that the scalar products «Ax ^{j},x^{j}» 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 language | English (US) |
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Pages (from-to) | 254-262 |

Number of pages | 9 |

Journal | Linear Algebra and Its Applications |

Volume | 444 |

DOIs | |

State | Published - Mar 1 2014 |

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### Keywords

- Almost normal matrices
- Numerical range
- Toeplitz matrices

### ASJC Scopus subject areas

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

### Cite this

*Linear Algebra and Its Applications*,

*444*, 254-262. https://doi.org/10.1016/j.laa.2013.11.045