On the span of Hadamard products of vectors

Let A1, … , Ak be positive semidefinite matrices and B1, … , Bk arbitrary complex matrices of order n. We show thatspan{(A1x)∘(A2x)∘⋯∘(Akx)|x∈Cn}=range(A1∘A2∘⋯∘Ak)span{(A1x)∘(A2x)∘⋯∘(Akx)|x∈Cn}=range(A1∘A2∘⋯∘Ak)andspan{(B1x1)∘(B2x2)∘⋯∘(Bkxk)|xj∈Cn}=range(B1B1∗)∘(B2B2∗)∘⋯∘(BkBk∗),where ∘ means the Hadamard product. This generalizes two recent results of Sun, Du and Liu.