TY - JOUR

T1 - Numerical ranges of companion matrices

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

N1 - Funding Information:
∗ Corresponding author. E-mail addresses: hlgau@math.ncu.edu.tw (H.-L. Gau), pywu@math.nctu.edu.tw (P.Y. Wu). 1 Part of the results here was presented by the second author in the 12th ILAS Conference at Regina, Canada in June 2005. The research was partially supported by the National Science Council of the Republic of China under projects NSC-94-2115-M-008-010 and NSC-94-2115-M-009-007, respectively.

PY - 2007/3/1

Y1 - 2007/3/1

N2 - We show that an n-by-n companion matrix A can have at most n line segments on the boundary ∂W(A) of its numerical range W(A), and it has exactly n line segments on ∂W(A) if and only if, for n odd, A is unitary, and, for n even, A is unitarily equivalent to the direct sum A1 ⊕ A2 of two (n/2)-by-(n/2) companion matricesA1 = fenced((0, 1; 0, {triple dot, diagonal NW-SE}; {triple dot, diagonal NW-SE}, 1; a, 0)) and A2 = fenced((0, 1; 0, {triple dot, diagonal NW-SE}; {triple dot, diagonal NW-SE}, 1; - 1 / over(a, ̄), 0)) with 1 ≤ {divides}a{divides} < tan(π/n) + sec(π/n).

AB - We show that an n-by-n companion matrix A can have at most n line segments on the boundary ∂W(A) of its numerical range W(A), and it has exactly n line segments on ∂W(A) if and only if, for n odd, A is unitary, and, for n even, A is unitarily equivalent to the direct sum A1 ⊕ A2 of two (n/2)-by-(n/2) companion matricesA1 = fenced((0, 1; 0, {triple dot, diagonal NW-SE}; {triple dot, diagonal NW-SE}, 1; a, 0)) and A2 = fenced((0, 1; 0, {triple dot, diagonal NW-SE}; {triple dot, diagonal NW-SE}, 1; - 1 / over(a, ̄), 0)) with 1 ≤ {divides}a{divides} < tan(π/n) + sec(π/n).

KW - Companion matrix

KW - Numerical range

UR - http://www.scopus.com/inward/record.url?scp=33846188598&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2006.03.037

DO - 10.1016/j.laa.2006.03.037

M3 - 期刊論文

AN - SCOPUS:33846188598

SN - 0024-3795

VL - 421

SP - 202

EP - 218

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 2-3 SPEC. ISS.

ER -