Zero-dilation index of a finite matrix ✩

  title={Zero-dilation index of a finite matrix ✩},
  author={Hwa-Long Gau and Kuo-Zhong Wang and Pei Yuan Wu},
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… CONTINUE READING