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http://hdl.handle.net/10397/6100
Title: | Higher rank numerical ranges of normal matrices | Authors: | Gau, HL Li, CK Poon, YT Sze, NS |
Issue Date: | 2011 | Source: | SIAM journal on matrix analysis and applications, 2011, v. 32, no. 1, p. 23-43 | 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. | Keywords: | Quantum error correction Higher rank numerical range Normal matrices Convex polygon |
Publisher: | Society for Industrial and Applied Mathematics | Journal: | SIAM journal on matrix analysis and applications | 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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Gau_Rank_Normal_Matrices.pdf | 9.49 MB | Adobe PDF | View/Open |
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