TY - JOUR

T1 - Crawford numbers of powers of a matrix

AU - Wang, Kuo Zhong

AU - Wu, Pei Yuan

AU - Gau, Hwa Long

N1 - Funding Information:
Corresponding author. E-mail addresses: kzwang@math.nctu.edu.tw (K.-Z. Wang), pywu@math.nctu.edu.tw (P.Y. Wu), hlgau@math.ncu.edu.tw (H.-L. Gau). 1 Research supported by a post-doctor fellowship of the National Science Council of the Republic of China. 2 Research supported by the National Science Council of the Republic of China under NSC 96-2115-M-009-013-MY3 and by the MOE-ATU project. 3 Research supported by the National Science Council of the Republic of China under NSC 98-2628-M-008-007.

PY - 2010/12/30

Y1 - 2010/12/30

N2 - or an n-by-n matrix A, its Crawford number c(A) (resp., generalized Crawford number C(A)) is, by definition, the distance from the origin to its numerical range W(A) (resp., the boundary of its numerical range ∂W(A)). It is shown that if A has eigenvalues λ1,⋯, λn arranged so that |λ 1|≥⋯≥|λn|, then lim kc(Ak)1/k (resp., limkC(A k)1/k) equals 0 or |λn| (resp., |λj| for some j, 1≤j≤n). For a normal A, more can be said, namely, lim kc(A(Ak)1/k=|λn| (resp., limkC((Ak)1/k=|λj| for some j, 3≤j≤n). In these cases, the above possible values can all be assumed by some A.

AB - or an n-by-n matrix A, its Crawford number c(A) (resp., generalized Crawford number C(A)) is, by definition, the distance from the origin to its numerical range W(A) (resp., the boundary of its numerical range ∂W(A)). It is shown that if A has eigenvalues λ1,⋯, λn arranged so that |λ 1|≥⋯≥|λn|, then lim kc(Ak)1/k (resp., limkC(A k)1/k) equals 0 or |λn| (resp., |λj| for some j, 1≤j≤n). For a normal A, more can be said, namely, lim kc(A(Ak)1/k=|λn| (resp., limkC((Ak)1/k=|λj| for some j, 3≤j≤n). In these cases, the above possible values can all be assumed by some A.

KW - Crawford number

KW - Generalized Crawford number

KW - Numerical range

UR - http://www.scopus.com/inward/record.url?scp=77957283000&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2010.08.004

DO - 10.1016/j.laa.2010.08.004

M3 - 期刊論文

AN - SCOPUS:77957283000

VL - 433

SP - 2243

EP - 2254

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 11-12

ER -