## Abstract

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 x_{j} such that the scalar products 〈Ax_{j},x_{j}〉 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 language | English (US) |
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Pages (from-to) | 192-218 |

Number of pages | 27 |

Journal | Linear Algebra and Its Applications |

Volume | 644 |

DOIs | |

State | Published - Jul 1 2022 |

## Keywords

- 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