## Abstract

An n-by-n (n ≥ 3) weighted shift matrix A is one of the form where the a j 's, called the weights of A, are complex numbers. Let A [j] denote the (n - 1)-by-(n - 1) principal submatrix of A obtained by deleting its jth row and jth column. We show that the boundary of numerical range W(A) has a line segment if and only if the a j 's are nonzero andW(A[k]) = W(A[l]) = W(A[m]) for some 1 ≤ k < l < m ≤ n. This refines previous results of Tsai and Wu on numerical ranges of weighted shift matrices. In addition, we give an example showing that there is a weighted shift matrix with line segments on the boundary of its numerical range such that the moduli of its weights are not periodic.

Original language | English |
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Pages (from-to) | 568-578 |

Number of pages | 11 |

Journal | Linear and Multilinear Algebra |

Volume | 62 |

Issue number | 5 |

DOIs | |

State | Published - May 2014 |

## Keywords

- numerical range
- weighted shift matrix