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## Abstract

The zero-dilation index d(A) of a square matrix A is the largest k for which A is unitarily similar to a matrix of the form (Formula presented), where 0_{k} denotes the k-by-k zero matrix. In this paper, it is shown that if A is an Sn-matrix or an n-by-n companion matrix, then d(A) is at most [n/2], the smallest integer greater than or equal to n/2. Those A’s for which the upper bound is attained are also characterized. Among other things, it is shown that, for an odd n, the S_{n}-matrix A is such that d(A) = (n+1)/2 if and only if A is unitarily similar to -A, and, for an even n, every n-by-n companion matrix A has d(A) equal to n/2.

Original language | English |
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Article number | 47 |

Pages (from-to) | 666-678 |

Number of pages | 13 |

Journal | Electronic Journal of Linear Algebra |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - 2016 |

## Keywords

- Companion matrix
- Numerical range
- S-matrix
- Zero-dilation index

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## Projects

- 1 Finished