Moment of a subspace and joint numerical range

Abstract

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