Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/6100
Title: Higher rank numerical ranges of normal matrices
Authors: Gau, HL
Li, CK
Poon, YT
Sze, NS 
Keywords: Quantum error correction
Higher rank numerical range
Normal matrices
Convex polygon
Issue Date: 2011
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on matrix analysis and applications, 2011, v. 32, no. 1, p. 23-43 How to cite?
Journal: SIAM journal on matrix analysis and applications 
Abstract: 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 ∈ M [sub n] has eigenvalues $a₁ ,...,a [sub n], then its higher rank numerical range A [sub k](A) is the intersection of convex polygons with vertices a [sub j1],...,a [sub j] [sub n-k+1], where $1 ≤ j [sub 1] <...< j [sub n-k+1]. 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 ∈ M [sub n] with minimum n such that A [sub k](A)=P. In particular, if P has p vertices, with p⋝3, there is a normal matrix A ∈ M [sub n] with n ≤ max{p+k-1,2k+2} such that A [sub k](A)=P.
URI: http://hdl.handle.net/10397/6100
ISSN: 0895-4798
EISSN: 1095-7162
DOI: 10.1137/09076430X
Rights: © 2011 Society for Industrial and Applied Mathematics
Appears in Collections:Journal/Magazine Article

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