TY - JOUR
T1 - Numerical ranges of companion matrices
AU - Gau, Hwa Long
AU - Wu, Pei Yuan
N1 - Funding Information:
∗ Corresponding author. E-mail addresses: [email protected] (H.-L. Gau), [email protected] (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 -