### 摘要

For an n-by-n complex matrix A, we define its zero-dilation index d(A) as the largest size of a zero matrix which can be dilated to A. This is the same as the maximum k (≥1) for which 0 is in the rank-k numerical range of A. Using a result of Li and Sze, we show that if d(A)> ⌊2n/3⌋, then, under unitary similarity, A has the zero matrix of size 3d(A)-2n as a direct summand. It complements the known fact that if d(A)> ⌊n/2⌋, then 0 is an eigenvalue of A. We then use it to give a complete characterization of n-by-n matrices A with d(A)=n-1, namely, A satisfies this condition if and only if it is unitarily similar to B⊕0_{n-3}, where B is a 3-by-3 matrix whose numerical range W(B) is an elliptic disc and whose eigenvalue other than the two foci of ∂W(B) is 0. We also determine the value of d(A) for any normal matrix A and any weighted permutation matrix A with zero diagonals.

原文 | English |
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頁（從 - 到） | 111-124 |

頁數 | 14 |

期刊 | Linear Algebra and Its Applications |

卷 | 440 |

發行號 | 1 |

DOIs | |

出版狀態 | Published - 1 一月 2014 |

## 指紋 深入研究「Zero-dilation index of a finite matrix」主題。共同形成了獨特的指紋。

## 引用此

*Linear Algebra and Its Applications*,

*440*(1), 111-124. https://doi.org/10.1016/j.laa.2013.10.041