Low Phase-Rank Approximation

In this paper, we propose and solve low phase-rank approximation problems, which serve as a counterpart to the well-known low-rank problem Schmidt-Mirsky theorem. It is well known that nonzero complex number can be specified by its gain phase, while it generally accepted gains of matrix may defined singular values, there no widely definition for phases. work, consider sectorial matrices, whose numerical ranges do not contain origin, adopt canonical angles such matrices their Similarly rank being define While associated with arithmetic mean, turns out natural parallel use geometric mean measure error. Importantly, derive majorization inequality between phases phases, similarly Ky-Fan eigenvalues Hermitian matrices. A characterization solutions proposed problem, same flavor theorem, then obtained in case where both objective approximant are restricted positive-imaginary. addition, provide an alternative formulation using geodesic distances The two formulations give rise exact set when involved additionally assumed unitary.