Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations
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- by Chi-Kwong Li and Nung-Sing Sze PDF
- Proc. Amer. Math. Soc. 136 (2008), 3013-3023 Request permission
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.References
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Additional Information
- Chi-Kwong Li
- Affiliation: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23185
- MR Author ID: 214513
- Email: ckli@math.wm.edu
- Nung-Sing Sze
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
- Email: sze@math.uconn.edu
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3013-3023
- MSC (2000): Primary 15A21, 15A24, 15A60, 15A90, 81P68
- DOI: https://doi.org/10.1090/S0002-9939-08-09536-1
- MathSciNet review: 2407062