Moment of a subspace and joint numerical range
Abel H. Klobouk, Alejandro Varela · Linear and Multilinear Algebra · 2022
For a subspace S of Cn and a fixed basis, we study the compact and convex set mS=convexhull {|s|2∈R≥0n:s∈S and ‖s‖=1}≃{Diag(Y)∈Mnh(C):Y≥0,tr(Y)=1,PSYPS=Y}that we call the moment of S, where |s|2=(|s1|2,|s2|2,…,|sn|2). This set is relevant in the determination of minimal hermitian matrices (M∈Mnh such that ‖M+D‖≤D for every diagonal D and the spectral norm ‖⋅‖). We describe extremal points and certain curves of mS in terms of principal vectors that minimize the angle between S and the coordinate axes of the fixed basis. We also relate mS to the joint numerical range W of n rank one n×n hermitian matrices constructed with orthogonal projection PS and the fixed basis {ei}i=1n used. This connection provides a new approach to the description of mS 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.