Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations
Chi-Kwong Li, Nung-Sing Sze · 2012
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 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 maximum dimension of a totally isotropic subspace of a square matrix, and verify the solvability of certain matrix equations. 1.