  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.
