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

## Abstract

Suppose A, D_{1},…,D_{m} are n × n matrices where A is self-adjoint, and let X = Σ^{m}_{k}=_{1}D_{k}AD*_{k}. It is shown that if ΣD_{k}D*_{k}=ΣD*_{k}D_{k} = I, then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D_{1},…,D_{m}, a result is obtained in terms of weak majorization. If each D_{k} 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 b_{ii} = 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.

Item Type: | Article |
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ID Code: | 1375 |

Deposited On: | 05 Oct 2010 12:40 |

Last Modified: | 31 Dec 2011 12:03 |

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