Moment of a subspace and joint numerical range

Abstract

For a subspace S of and a fixed basis, we study the compact and convex set that we call the moment of S, where . This set is relevant in the determination of minimal hermitian matrices ( such that for every diagonal D and the spectral norm ). We describe extremal points and certain curves of in terms of principal vectors that minimize the angle between S and the coordinate axes of the fixed basis. We also relate to the joint numerical range W of n rank one hermitian matrices constructed with orthogonal projection and the fixed basis used. This connection provides a new approach to the description of and to minimal matrices. As a consequence, the intersection of two of these joint numerical ranges corresponding to orthogonal subspaces allows the construction or detection of a minimal matrix.