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

TL;DR

This paper characterizes the higher rank numerical range of matrices, showing its convexity and polygonal structure for normal matrices, and applies these results to quantum error correction, isotropic subspaces, and matrix equations.

Contribution

It provides a complete description of the higher rank numerical range using matrix canonical forms, confirming conjectures and deriving bounds relevant to quantum information and matrix theory.

Findings

Higher rank numerical range is convex and can be represented as an intersection of half planes.

For normal matrices, the numerical range is a convex polygon determined by eigenvalues.

Derived an upper bound for the dimension of totally isotropic subspaces and verified matrix equation solvability.

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.