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

Abstract

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.

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

References in corpus (2)

• Canonical forms for complex matrix congruence and *congruence
• Quantum Error Correcting Codes From The Compression Formalism

Cited by in corpus (13)

• Condition for the higher rank numerical range to be non-empty
• Higher rank numerical ranges and low rank perturbations of quantum channels
• Recovery in quantum error correction for general noise without measurement
• Efficient Quantum Error Correction for Fully Correlated Noise
• Numerical shadows: measures and densities on the numerical range
• Higher rank numerical ranges of normal matrices
• Higher-rank Numerical Ranges and Kippenhahn Polynomials
• Isospectrality and matrices with concentric circular higher rank numerical ranges
• Higher Rank Numerical Ranges of Normal Operators and unitary dilations
• Higher Rank Numerical Ranges and Unitary Dilations
• On the boundary of weighted numerical ranges
• On the higher rank numerical range of the shift
• On the ellipticity of the higher rank numerical range