Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations

Details Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations

2007-06-11

arxiv.org/abs/0706.1536v2

Proc. Amer. Math. Soc., 136:3013-3023. 2008

Mathematics

Functional Analysis

10 pages. To appear in Proceedings of the American Mathematical Society

Scientific paper

10.1090/S0002-9939-08-09536-1

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 $\mathcal C$. 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 verify the solvability of certain matrix equations.

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