We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary matrix are given by complex polygons determined by the spectral structure of the matrix. We discuss applications of the results to quantum error correction, s...

In most primal-dual interior point methods, a sequence of weighted normal equations are solved, where the only change from one problem to the next are the weights and the right hand side. Solving the normal equations alternating between a direct method and an iterative method was introduced in Wang and O'Leary 11]. A class of preconditioners based on a low-rank correction of a Cholesky factoriz...

A group $G$ has a finite special rank $r$ if every finitely generated subgroup of is by at most elements and there which exactly generators. If not such $r$, then we say that infinite rank. In this paper, study generalized radical non-abelian groups whose subgroups are transitively normal.