On majorization and Schur products

Bapat, R. B. ; Sunder, V. S. (1985) On majorization and Schur products Linear Algebra and its Applications, 72 . pp. 107-117. ISSN 0024-3795

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/002437...

Related URL: http://dx.doi.org/10.1016/0024-3795(85)90147-8


Suppose 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.

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