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

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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2008-09, Vol.136 (9), p.3013-3023
Main Authors: Li, Chi-Kwong, Sze, Nung-Sing
Format: Article
Language:English
Subjects:
Algebra
Canonical forms
Convex and discrete geometry
Convexity
Eigenvalues
Exact sciences and technology
Finite differences and functional equations
General mathematics
General, history and biography
Geometry
Half planes
Linear algebra
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematical theorems
Mathematics
Matrices
Matrix equations
Sciences and techniques of general use
Solvability
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-08-09536-1