TY - JOUR

T1 - Higher rank numerical ranges of normal matrices

AU - Gau, Hwa Long

AU - Li, Chi Kwong

AU - Poon, Yiu Tung

AU - Sze, Nung Sing

PY - 2011

Y1 - 2011

N2 - The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ε Mn has eigenvalues a1, ⋯ , a n, then its higher rank numerical range Γκ(A) is the intersection of convex polygons with vertices aj1 , ⋯ , ajn-k+1, where 1 ≤ j1 < ⋯ ≤ j n-k+1 ≤ n. In this paper, it is shown that the higher rank numerical range of a normal matrix with m distinct eigenvalues can be written as the intersection of no more than max{m, 4} closed half planes. In addition, given a convex polygon P, a construction is given for a normal matrix A ε Mn with minimum n such that Δκ(A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ε Mn with n ≤ max {p + k -1, 2k + 2} such that Γκ(A) = P.

AB - The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ε Mn has eigenvalues a1, ⋯ , a n, then its higher rank numerical range Γκ(A) is the intersection of convex polygons with vertices aj1 , ⋯ , ajn-k+1, where 1 ≤ j1 < ⋯ ≤ j n-k+1 ≤ n. In this paper, it is shown that the higher rank numerical range of a normal matrix with m distinct eigenvalues can be written as the intersection of no more than max{m, 4} closed half planes. In addition, given a convex polygon P, a construction is given for a normal matrix A ε Mn with minimum n such that Δκ(A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ε Mn with n ≤ max {p + k -1, 2k + 2} such that Γκ(A) = P.

KW - Convex polygon

KW - Higher rank numerical range

KW - Normal matrices

KW - Quantum error correction

UR - http://www.scopus.com/inward/record.url?scp=79952430162&partnerID=8YFLogxK

U2 - 10.1137/09076430X

DO - 10.1137/09076430X

M3 - 期刊論文

AN - SCOPUS:79952430162

SN - 0895-4798

VL - 32

SP - 23

EP - 43

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 1

ER -