On majorization and Schur products

AbstractSuppose A, D1,…,Dm are n × n matrices where A is self-adjoint, and let X = Σmk = 1DkAD∗k. It is shown that if ΣDkD∗k = ΣD∗ kDk = I, then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D1,…,Dm, a result is obtained in terms of weak majorization. If each Dk is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if bii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved