On Eigenvalue and Singular Value Inequalities for Matrix Product

Let H∈Cn×n have real eigenvalues λ1(H)≥⋯≥λn(H). It is known that if G and H are two nonnegative matrices, then ∑kt=1λt(GH)≥∑kt=1λt(G)λn−t+1(H). The authors prove that in this case if 1≤i1 ∑t=1kλit(GH)≥∑t=1kλit(G)λn−t+1(H) and ∑t=1kλt(GH)≥∑t=1kλit(G)λn−it+1(H)