An eigenvalue majorization inequality for positive semidefinite block matrices

摘要

Let be a Hermitian matrix. It is known that the vector of diagonal elements of H, diag(H), is majorized by the vector of the eigenvalues of H, λ(H), and that this majorization can be extended to the eigenvalues of diagonal blocks of H. Reverse majorization results for the eigenvalues are our goal. Under the additional assumptions that H is positive semidefinite and the block K is Hermitian, the main result of this article provides a reverse majorization inequality for the eigenvalues. This results in the following majorization inequalities when combined with known majorization inequalites on the left: diag(H) < λ(M ? N) < λ(H) < λ ((M + N) ? 0).