|
Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations
Author(s):
Chi-Kwong
Li;
Nung-Sing
Sze
Journal:
Proc. Amer. Math. Soc.
136
(2008),
3013-3023.
MSC (2000):
Primary 15A21, 15A24, 15A60, 15A90, 81P68
Posted:
April 30, 2008
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
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  . 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:
-
- 1.
- M.-D. Choi, M. Giesinger, J. A. Holbrook, and D. W. Kribs, Geometry of higher-rank numerical ranges, Linear and Multilinear Algebra 56 (2008), 53-64. MR 2378301
- 2.
- M.-D. Choi, J. A. Holbrook, D. W. Kribs, and K. Życzkowski, Higher-rank numerical ranges of unitary and normal matrices, Operators and Matrices 1 (2007), 409-426. MR 2344684
- 3.
- M.-D. Choi, D. W. Kribs, and K. Życzkowski, Higher-rank numerical ranges and compression problems, Linear Algebra Appl. 418 (2006), 828-839. MR 2260232 (2007k:15041)
- 4.
- M.-D. Choi, D. W. Kribs, and K. Życzkowski, Quantum error correcting codes from the compression formalism, Rep. Math. Phys. 58 (2006), 77-91. MR 2273568 (2007h:81039)
- 5.
- K. Fan and G. Pall, Imbedding conditions for Hermitian and normal matrices, Canad. J. Math. 9 (1957), 298-304. MR 0085216 (19:6e)
- 6.
- R. A. Horn and V. V. Sergeichuk, Canonical forms for complex matrix congruence and
 -congruence, Linear Algebra Appl. 416 (2006), 1010-1032. MR 2242477 (2007c:15018) - 7.
- P. Lancaster and L. Rodman, Algebraic Riccati equations, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 1367089 (97b:93003)
- 8.
- C.-K. Li, A simple proof of the elliptical range theorem, Proc. Amer. Math. Soc. 124 (1996), 1985-1986. MR 1322932 (96i:15026)
- 9.
- C.-K. Li, Y. T. Poon and N.-S. Sze, Condition for the higher rank numerical range to be non-empty, Linear and Multilinear Algebra, to appear.
- 10.
- C.-K. Li, Y. T. Poon and N.-S. Sze, Higher rank numerical ranges and low rank perturbations of quantum channels, preprint. http://arxiv.org/abs/0710.2898
- 11.
- G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990. MR 1061154 (92a:65017)
- 12.
- H. Woerdeman, The higher rank numerical range is convex, Linear and Multilinear Algebra 56 (2008), 65-67. MR 2378302
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
15A21, 15A24, 15A60, 15A90, 81P68
Retrieve articles in all Journals with MSC
(2000):
15A21, 15A24, 15A60, 15A90, 81P68
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:
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
Posted:
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 of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org