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

Li, Chi-Kwong; Sze, Nung-Sing

doi:10.1090/S0002-9939-08-09536-1

Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations
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by Chi-Kwong Li and Nung-Sing Sze

Proc. Amer. Math. Soc. 136 (2008), 3013-3023

DOI: https://doi.org/10.1090/S0002-9939-08-09536-1

Published electronically: April 30, 2008

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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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