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Generalizations of two inequalities involving hermitian forms
Authors:Sin-Chung Chang
Institution:National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135, USA
Abstract:Let λ1 and λN be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(hij), and x=(x1,x2,…,xN) with (x,x)=1. Then, it is known that (1) λ1?(x,Hx)?λN and (2) if, in addition, H is positive definite, 1N)21λN?(x,Hx)(x,H?1x)?1. Assuming that y=(y1,y2,…, yN) and |yi|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H?1 are, respectively, replaced by the Hadamard products M(y)1H and M(y)1H?1, where M(y) is a matrix defined by M(y)=(δij+(1?δij)yiyj. Subsequently, these results are extended to improve the spectral bounds of M(y)1H.
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