### Abstract

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.

Original language | English |
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Pages (from-to) | 111-124 |

Number of pages | 14 |

Journal | Linear Algebra and Its Applications |

Volume | 440 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2014 |

### Keywords

- Higher-rank numerical range
- Normal matrix
- Weighted permutation matrix
- Zero-dilation index

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## Cite this

*Linear Algebra and Its Applications*,

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