TY - JOUR
T1 - Proof of a conjecture on numerical ranges of weighted cyclic matrices
AU - Gau, Hwa Long
N1 - Publisher Copyright:
© 2023 Elsevier Inc.
PY - 2024/2/1
Y1 - 2024/2/1
N2 - Recall that the n-by-n weighted cyclic matrix with weights a1,…,an(∈C) is the matrix C(a1,…,an)=[0a10⋱⋱an−1an0], and W(C(a1,…,an)) is the numerical range of C(a1,…,an). Let Sn be the symmetric group on {1,…,n}. In [2], Chien et al. conjecture that if |a1|≥|a2|≥…≥|an| then W(C(aη(1),…,aη(n)))⊆W(C(aσn(1),…,aσn(n))) for any permutation η∈Sn, where σn∈Sn is defined by (σn(1),…,σn(n))={(n−1,…,4,2,1,3,5,…,n−2,n)if n is odd,(n−2,…,4,2,1,3,5,…,n−1,n)if n is even. In this note, we settle the conjecture in the affirmative.
AB - Recall that the n-by-n weighted cyclic matrix with weights a1,…,an(∈C) is the matrix C(a1,…,an)=[0a10⋱⋱an−1an0], and W(C(a1,…,an)) is the numerical range of C(a1,…,an). Let Sn be the symmetric group on {1,…,n}. In [2], Chien et al. conjecture that if |a1|≥|a2|≥…≥|an| then W(C(aη(1),…,aη(n)))⊆W(C(aσn(1),…,aσn(n))) for any permutation η∈Sn, where σn∈Sn is defined by (σn(1),…,σn(n))={(n−1,…,4,2,1,3,5,…,n−2,n)if n is odd,(n−2,…,4,2,1,3,5,…,n−1,n)if n is even. In this note, we settle the conjecture in the affirmative.
KW - Numerical radius
KW - Numerical range
KW - Weighted cyclic matrix
UR - http://www.scopus.com/inward/record.url?scp=85178078465&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2023.11.017
DO - 10.1016/j.laa.2023.11.017
M3 - 期刊論文
AN - SCOPUS:85178078465
SN - 0024-3795
VL - 682
SP - 295
EP - 308
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
ER -