Moment of a subspace and joint numerical range

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