Zero-dilation index of a finite matrix

Hwa Long Gau, Kuo Zhong Wang, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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⊕0n-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 languageEnglish
Pages (from-to)111-124
Number of pages14
JournalLinear Algebra and Its Applications
Issue number1
StatePublished - 1 Jan 2014


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


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