TY - JOUR

T1 - Zero-dilation index of a finite matrix

AU - Gau, Hwa Long

AU - Wang, Kuo Zhong

AU - Wu, Pei Yuan

N1 - Funding Information:
Research supported by the National Science Council of the Republic of China under the NSC-101-2115-M-008-006 , NSC-101-2115-M-009-001 and NSC-101-2115-M-009-004 projects, respectively. The third author also acknowledges the support from the MOE-ATU project.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Higher-rank numerical range

KW - Normal matrix

KW - Weighted permutation matrix

KW - Zero-dilation index

UR - http://www.scopus.com/inward/record.url?scp=84889886612&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2013.10.041

DO - 10.1016/j.laa.2013.10.041

M3 - 期刊論文

AN - SCOPUS:84889886612

SN - 0024-3795

VL - 440

SP - 111

EP - 124

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1

ER -