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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations


Authors: Chi-Kwong Li and Nung-Sing Sze
Journal: Proc. Amer. Math. Soc. 136 (2008), 3013-3023
MSC (2000): Primary 15A21, 15A24, 15A60, 15A90, 81P68
Published electronically: April 30, 2008
MathSciNet review: 2407062
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Abstract: The results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in $ \mathbb{C}$. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix and to verify the solvability of certain matrix equations.


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Additional Information

Chi-Kwong Li
Affiliation: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23185
Email: ckli@math.wm.edu

Nung-Sing Sze
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: sze@math.uconn.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09536-1
PII: S 0002-9939(08)09536-1
Keywords: Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations.
Received by editor(s): March 26, 2007
Published electronically: April 30, 2008
Additional Notes: The research of Li was partially supported by an NSF grant and an HK RGC grant. He is an honorary professor of the University of Hong Kong.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.