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 -