Some Inequalities for the Eigenvalues of the Product of Positive Semi-definite Hermitian Matrices

Let λ1(A)⩾⋯⩾λn(A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽i1\u3c ⋯ \u3cik⩽n. Our main results are ∑t=1kλt(GH)⩽∑t=1kλit(G)λn−it+1(H) And ∑t=1kλit(GH)⩽∑t=1kλit(G)λn−t+1(H) Here G and H are n by n positive semidefinite Hermitian matrices. These results extend Marshall and Olkin\u27s inequality ∑t=1kλt(GH)⩽∑t=1kλt(G)λn−t+1(H) We also present analogous results for singular values